user672: 
While posting this question, I stumbled on this list of variants from Wikipedia, which I must admit is totally awesome. 
I know a few online servers allow a few variants, but am not aware of places where many variants players would gather, which would they be ?
Are there famous clubs for on the board variants play ?
Collections of fairy chess problems ? 
This list of fairy pieces is not at all bad, but could use a few more easy-to-understand diagrams, and some art for representations.
I guess there's enough to say on this topic for books to be published on the subject ?


user4578: Try Connect Score. The game combines Chess with Dots and Boxes. The game can be viewed at: http://connect-score.blogspot.ca/
Rick Nordal

user672: There's a training site on variants : https://chessvariants.training/

We support these variants: Antichess, Atomic chess, King of the Hill, Three-check, Horde chess and Racing Kings.

So more or less the set of variants supported on lichess. It's mostly puzzles about tactics, and some about endgames. 

user4215: There is now a theory book on Horde Chess, full with annotated games and strategies.

user3158: As I have seen it, FICS seems to be the best place for variants. But number of people playing all variants vary. The following are supported on FICS-
Chess
BugHouse
CrazyHouse
Suicide
Atomic
FischerRandom
Losers
Wild 0 through Wild 8(a)


user3456: You can play correspondence chess variants on the Apronus.com interactive chessboard by generating links to click which encode the board position and all the mvoes made.
For example
a short game on an 8x5 board.

(source: apronus.com)

user15989: Apart from the main chess sites (lichess.org, chess.com), there are a number of other sites where variants can be played:
Pychess-variants:
Based on lichess's source code and powered by Fairy-Stockfish. The list of supported variants is huge, including popular ones like S-Chess, Capablanca, and crazyhouse and 960 variants of these, as well as xiangqi, janggi, shogi, makruk, sittuyin, and even variants thereof. You can play against humans or challenge different levels of Fairy-Stockfish, which is the strongest available AI for many of the variants offered. The userbase is still growing (circa June 2020), but the most reliable method for now is to ask in the lobby or in the Pychess Discord server for a game.
vchess: Still under constant development, but offers a huge amount of western chess variants, including many not offered elsewhere. There are also options for normal, symmetrical or asymmetrical 960 starting positions for many variants. Best to make appointments with other variant enthusiasts to meet here and play.
PlayOK: An established site with a large userbase. Aside from the big regional variants of chess, xiangqi, makruk and shogi, also features many other board and card games. Many Asian players play xiangqi and makruk here.

As for chess variant communities, with the rise of new social apps and the appearance of variants on the major modern chess sites, the previously disparate communities started to coalesce and band together.
PyChess Players Discord: As previously mentioned, the Pychess site has an associated Discord. Channels for individual variants exist, as well as being the place to ask anyone online for games. The developers, translators, and other volunteers also hang out here. The community is still growing as the site improves and becomes more well known.
The House Discord: One of the largest (and strongest?) chess variant communities. A Discord server bringing together players of all strengths and many variants, principally those offered on lichess.org, chess.com, and Pychess. Many of the best players in the world in each variant hang out here (this is not hyperbole). These players have also created valuable resources to help others learn and improve in the variants. Definitely worth checking out if you're thinking of improving in a variant.

user3819: I am surprised that no one yet mentioned http://www.chessvariants.com/ --- the most comprehensive web site on Chess variants I am aware of. They also support a play-by-mail server for many Chess variants.
The Chess variants offer there are range from Modest variants (which is probably close to the variants the original poster is looking for) to rather wild and deviating variant.

user884: I love playing chess variants. For playing live games, FICS supports the major chess variants: Suicide, Atomic, Losers, Crazyhouse and Upside-Down. 
For playing correspondence games, I use both Scheming Mind and BrainKing. They each have the normal variants (above), as well as a few unique ones. The ones unique to Scheming Mind include Benedict Chess and Chicken Chess. The ones unique to BrainKing include Ice Age Chess and Cheshire Cat Chess. 
These sites are free, for a limited number of games. There are ladders and tournaments, so you can play against other people who enjoy the same variants. You can find me there under my real name (I do not use a handle).

user780: Unfortunately I don't know of a chess server where they play chess variants, (see update below) but I have a few other resources.  
The program Winboard has several variants, including Shogi, Xiangqi, Shatranj and Atomic.  
I also found a very cool blog, the guy there goes over a lot of chess and checker variants.  Its pretty cool!
Here is a article he has about Fairy Chess.
There are also several books written on the subject including:

Chess Variants & Games  
Popular Chess Variants

And the Encyclopedia of  Chess Variants 
Update
In my ramblings I ran accross this page, indicating that the ICC allows chess variants.
Quote:  

ICC offers a number of chess variants.  

I also found a book on Fairy Chess. Unfortunately it appears to be out of print.  

672: <ul>
<li><p>While posting this question, I stumbled on <a href="https://chess.stackexchange.com/a/214/672">this <strong>list of variants</strong></a> <a href="http://en.wikipedia.org/wiki/Chess_variant" rel="noreferrer">from Wikipedia</a>, which I must admit is totally awesome. </p></li>
<li><p>I know a few <strong>online servers</strong> allow a few variants, but am not aware of places where <em>many</em> variants players would gather, which would they be ?</p></li>
<li><p>Are there famous clubs for <strong>on the board</strong> variants play ?</p></li>
<li><p>Collections of <a href="http://en.wikipedia.org/wiki/Fairy_chess" rel="noreferrer">fairy chess</a> <strong>problems</strong> ? </p></li>
<li><p>This <a href="http://en.wikipedia.org/wiki/Fairy_chess_piece" rel="noreferrer">list of <strong>fairy pieces</strong></a> is not at all bad, but could use a few more easy-to-understand diagrams, and some art for representations.</p></li>
<li><p>I guess there's enough to say on this topic for <strong>books</strong> to be published on the subject ?</p></li>
</ul>
 4578: <p>Try Connect Score. The game combines Chess with Dots and Boxes. The game can be viewed at: <a href="http://connect-score.blogspot.ca/" rel="nofollow">http://connect-score.blogspot.ca/</a></p>

<p>Rick Nordal</p>
 672: <p>There's a training site on variants : <a href="https://chessvariants.training/" rel="nofollow noreferrer">https://chessvariants.training/</a></p>

<blockquote>
  <p>We support these variants: Antichess, Atomic chess, King of the Hill, Three-check, Horde chess and Racing Kings.</p>
</blockquote>

<p>So more or less the set of variants supported on lichess. It's mostly puzzles about tactics, and some about endgames. </p>
 4215: <p>There is now a <a href="https://rads.stackoverflow.com/amzn/click/com/B092XPNNVS" rel="nofollow noreferrer" rel="nofollow noreferrer">theory book</a> on Horde Chess, full with annotated games and strategies.</p>
 3158: <p>As I have seen it, FICS seems to be the best place for variants. But number of people playing all variants vary. The following are supported on FICS-</p>

<pre><code>Chess
BugHouse
CrazyHouse
Suicide
Atomic
FischerRandom
Losers
Wild 0 through Wild 8(a)
</code></pre>
 3456: <p>You can play correspondence chess variants on the Apronus.com interactive chessboard by generating links to click which encode the board position and all the mvoes made.</p>
<p>For example
<a href="https://www.apronus.com/chess/pgnviewer/?m=Nf3*_gxf3_gxf3_exf3_exf3_Nxf3*_Ke2_Qe4M&amp;h=5" rel="nofollow noreferrer">a short game on an 8x5 board</a>.</p>
<p><a href="https://i.stack.imgur.com/MqmhG.gif" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/MqmhG.gif" alt="animated diagram" /></a><br />
<sub>(source: <a href="http://www.apronus.com/chess/moviediagram.php?a=rnbqkbnrHppppppppH8HPPPPPPPPHRNBQKBNR_rnbqkbnrHppppppppH5N2HPPPPPPPPHRNBQKB1R_rnbqkbnrHpppppp1pH5p2HPPPPPPPPHRNBQKB1R_rnbqkbnrHpppppp1pH5P2HPPPPPP1PHRNBQKB1R_rnbqkbnrHpppp1p1pH5p2HPPPPPP1PHRNBQKB1R_rnbqkbnrHpppp1p1pH5P2HPPPP1P1PHRNBQKB1R_rnbqkb1rHpppp1p1pH5n2HPPPP1P1PHRNBQKB1R_rnbqkb1rHpppp1p1pH5n2HPPPPKP1PHRNBQ1B1R_rnb1kb1rHppppqp1pH5n2HPPPPKP1PHRNBQ1B1R&amp;d=MRNBQ_B_RPPPPKP_P_____n__ppppqp_prnb_kb_r0&amp;w=8&amp;h=5/" rel="nofollow noreferrer">apronus.com</a>)</sub></p>
 15989: <p>Apart from the main chess sites (lichess.org, chess.com), there are a number of other sites where variants can be played:</p>

<p><strong><a href="https://www.pychess.org" rel="noreferrer">Pychess-variants</a></strong>:
Based on lichess's source code and powered by Fairy-Stockfish. The <a href="https://www.pychess.org/variant" rel="noreferrer">list of supported variants</a> is huge, including popular ones like S-Chess, Capablanca, and crazyhouse and 960 variants of these, as well as xiangqi, janggi, shogi, makruk, sittuyin, and even variants thereof. You can play against humans or challenge different levels of Fairy-Stockfish, which is the strongest available AI for many of the variants offered. The userbase is still growing (circa June 2020), but the most reliable method for now is to ask in the lobby or in the Pychess Discord server for a game.</p>

<p><strong><a href="https://vchess.club" rel="noreferrer">vchess</a></strong>: Still under constant development, but offers a huge amount of western chess variants, including many not offered elsewhere. There are also options for normal, symmetrical or asymmetrical 960 starting positions for many variants. Best to make appointments with other variant enthusiasts to meet here and play.</p>

<p><strong><a href="https://www.playok.com" rel="noreferrer">PlayOK</a></strong>: An established site with a large userbase. Aside from the big regional variants of chess, xiangqi, makruk and shogi, also features many other board and card games. Many Asian players play xiangqi and makruk here.</p>

<hr>

<p>As for chess variant communities, with the rise of new social apps and the appearance of variants on the major modern chess sites, the previously disparate communities started to coalesce and band together.</p>

<p><strong><a href="https://discord.gg/aPs8RKr" rel="noreferrer">PyChess Players Discord</a></strong>: As previously mentioned, the Pychess site has an associated Discord. Channels for individual variants exist, as well as being the place to ask anyone online for games. The developers, translators, and other volunteers also hang out here. The community is still growing as the site improves and becomes more well known.</p>

<p><strong><a href="https://dscrd.me/housechessvariants" rel="noreferrer">The House Discord</a></strong>: One of the largest (and strongest?) chess variant communities. A Discord server bringing together players of all strengths and many variants, principally those offered on lichess.org, chess.com, and Pychess. Many of the best players in the world in each variant hang out here (this is not hyperbole). These players have also created valuable resources to help others learn and improve in the variants. Definitely worth checking out if you're thinking of improving in a variant.</p>
 3819: <p>I am surprised that no one yet mentioned <a href="http://www.chessvariants.com/" rel="nofollow noreferrer">http://www.chessvariants.com/</a> --- the most comprehensive web site on Chess variants I am aware of. They also support a play-by-mail server for many Chess variants.</p>

<p>The Chess variants offer there are range from <a href="https://chessvariants.com/index/mainquery.php?category=Usual-Modest" rel="nofollow noreferrer">Modest variants</a> (which is probably close to the variants the original poster is looking for) to rather wild and deviating variant.</p>
 884: <p>I love playing chess variants. For playing live games, FICS supports the major chess variants: Suicide, Atomic, Losers, Crazyhouse and Upside-Down. </p>

<p>For playing correspondence games, I use both <a href="http://schemingmind.com">Scheming Mind</a> and <a href="http://brainking.com">BrainKing</a>. They each have the normal variants (above), as well as a few unique ones. The ones unique to Scheming Mind include Benedict Chess and Chicken Chess. The ones unique to BrainKing include Ice Age Chess and Cheshire Cat Chess. </p>

<p>These sites are free, for a limited number of games. There are ladders and tournaments, so you can play against other people who enjoy the same variants. You can find me there under my real name (I do not use a handle).</p>
 780: <p><del>Unfortunately I don't know of a chess server where they play chess variants,</del> (see update below) but I have a few other resources.  </p>

<p>The program <a href="http://www.gnu.org/software/xboard/#tag-A1" rel="noreferrer">Winboard</a> has several variants, including Shogi, Xiangqi, Shatranj and Atomic.  </p>

<p>I also found a very cool <a href="http://abstractstrategygames.blogspot.com/" rel="noreferrer">blog</a>, the guy there goes over a lot of chess and checker variants.  Its pretty cool!<br>
<a href="http://abstractstrategygames.blogspot.com/2010/10/fairy-chess.html" rel="noreferrer">Here</a> is a article he has about Fairy Chess.</p>

<p>There are also several books written on the subject including:</p>

<ul>
<li><a href="http://rads.stackoverflow.com/amzn/click/9381115745" rel="noreferrer">Chess Variants &amp; Games </a>  </li>
<li><a href="http://rads.stackoverflow.com/amzn/click/0713485787" rel="noreferrer">Popular Chess Variants</a></li>
</ul>

<p>And the <a href="http://rads.stackoverflow.com/amzn/click/0952414201" rel="noreferrer">Encyclopedia of  Chess Variants</a> </p>

<p><em><strong>Update</em></strong><br>
In my ramblings I ran accross <a href="http://www6.chessclub.com/helpcenter/tips/wild.html" rel="noreferrer">this</a> page, indicating that the ICC allows chess variants.<br>
Quote:  </p>

<blockquote>
  <p>ICC offers a number of chess variants.  </p>
</blockquote>

<p>I also found a book on <a href="http://rads.stackoverflow.com/amzn/click/0486226875" rel="noreferrer">Fairy Chess</a>. Unfortunately it appears to be out of print.  </p>
