user725: I can't play chess very well, and want a computer game to play against. I've tried two so far - Chessbase (Fritz 5.32) and Crafty Chess. Both had the same problem that even on the lowest difficulty setting, the computer trashed me.
I'm running Windows XP. Here are my criteria:

100% Free (no nagware)
Lightweight: loads quickly, small filesize (say under 15 Mb)
Not system intensive. I don't need or want a 3d board.
Multiple difficulty levels, the lowest one being comparable to someone who's just been told what the legal moves are, and the highest one being... good. Good enough that I won't have to switch to a more advanced program for a while.

If it could have a nice clean interface without clutter everywhere, like the current move list and advanced AI options and what not - which might be great for advanced players but I have no need for - that would be nice.
If you know of a program like this for other operating systems, go ahead and post them for future readers... but I'm on XP.

user3852: Download Chess Wizard from this site:
http://www.chess-wizard.com/
It's completely free and lightweight. It also supports UCI engines. You can download the BrutusRND engine, which plays random legal moves.
You can also download ufim_802, which is a very weak engine.

user725: I'd like to answer my own question with a recommendation for Chessology.
Interface is pretty clean, and good difficulty settings, I found.
It also has an interesting feature: it colors squares of the board to reflect which player controls them. Red for black control, green for white control, with the color varying along a spectrum to represent weaker or stronger control (I think the formula is just Control = WhitePiecesThreateningSquare - BlackPiecesThreateningSquare, but I haven't done extensive testing). How useful that feature is I don't know because I don't know much about Chess, I'll leave it up to you esteemed gentlemen to discuss in the comments.

user3456: You can try the online option at https://www.apronus.com/chess/playcomputer/. You can experiment with the strength of the computer by reducing its time to think per move (in milliseconds).

user884: Another free chess program under Windows is Zillions of Games. Zillions is a generalized game system that includes chess as well as other games. You can easily control its intelligence. There are different settings for
1. Skill level
2. How variable its move choices are
3. How fast the computer moves   
I purchased the full version of Zillions, because I wanted to play more games than those included in the free download version.

user780: I find that SparkChess is fast and easy to beat on its lowest level.  It has a neat flash based interface, though you'll need Adobe's Flash Plugin to play.

user426: My favourite chess engine is Numpty chess.
I am an "average"(?!) rated chess player and I have had hours of fun playing against this engine and look forward to more. It is not so highly rated like some other engines eg StockFish, Rybka, Fritz etc etc and is great for learning & improving at chess. This means you do get winning chances and so you do not get so demoralised as when playing super strength engines. 
It needs a GUI to run in & is easily setup in Winboard or Arena chess gui. Chris Tatham is the engine author and a fine friend of mine (only through Numpty!).
Numpty is at Numpty chess engine and is freeware. Give it a game & enjoy!

user35: If I'm not on my own computer, I use Shredder Chess Online. You probably can't get much simpler than that. Unfortunately it has just 3 levels, but fortunately the middle level fits for me if I don't want to think too hard, and the highest level if I'm in serious mood :-)

user721: Lucas Chess is great! From the description of the program:

"The computer uses different chess programs (so-called chess engines) of various strength. The user starts playing against the weakest engine at first. Initially the engine plays with limited strength but as the user wins more games the engine will be given more calculation time and its strength will improve. Eventually the engine will reach its maximum level of strength and if the user continues to win he will be passed to the next stronger engine and so forth.

It comes with a tutor that watches over your moves.
Lucas Chess ships with several chess engines from very weak to very strong (Stockfish,Rybka) and thousands of training positions (mate in 2,3,4 .. etc.)
For improving your openings try ChessHero or ChessPositionTrainer

user731: You could try WinBoard. WinBoard is a GUI, that you can use with thousands of engines. Two of the most famously weak engines, that you can use it with it are:

Ufim 0.82 The limit strength can be set as low as 700 ELO
BrutusRND This one literally makes random (legal) moves. 

Source: chessforums
When you outgrow those engines, it will be easy to switch to a stronger one. The advantage of having multiple Engines is, that they all play different and so you won't get bored as fast.
Additionally, you could also weaken any existing engine. Here is an example with Crafty.

user745: Download Babaschess, sign up for the Free Internet Chess Server (FICS) and start playing unrated games against real people. :)

725: <p>I can't play chess very well, and want a computer game to play against. I've tried two so far - Chessbase (Fritz 5.32) and Crafty Chess. Both had the same problem that even on the lowest difficulty setting, the computer trashed me.</p>

<p>I'm running Windows XP. Here are my criteria:</p>

<ul>
<li>100% Free (no nagware)</li>
<li>Lightweight: loads quickly, small filesize (say under 15 Mb)</li>
<li>Not system intensive. I don't need or want a 3d board.</li>
<li>Multiple difficulty levels, the lowest one being comparable to someone who's just been told what the legal moves are, and the highest one being... good. Good enough that I won't have to switch to a more advanced program for a while.</li>
</ul>

<p>If it could have a nice clean interface without clutter everywhere, like the current move list and advanced AI options and what not - which might be great for advanced players but I have no need for - that would be nice.</p>

<p>If you know of a program like this for other operating systems, go ahead and post them for future readers... but I'm on XP.</p>
 3852: <p>Download Chess Wizard from this site:</p>

<p><a href="http://www.chess-wizard.com/" rel="nofollow noreferrer">http://www.chess-wizard.com/</a></p>

<p>It's completely free and lightweight. It also supports UCI engines. You can <a href="http://www.xs4all.nl/~vermeire/BrutusRND.rar" rel="nofollow noreferrer">download the BrutusRND engine</a>, which plays random legal moves.</p>

<p>You can also <a href="http://wbec-ridderkerk.nl/html/details1/Ufim.html" rel="nofollow noreferrer">download ufim_802</a>, which is a very weak engine.</p>
 725: <p>I'd like to answer my own question with a recommendation for <a href="https://chessology-sampler.software.informer.com/" rel="nofollow noreferrer">Chessology</a>.</p>

<p>Interface is pretty clean, and good difficulty settings, I found.</p>

<p>It also has an interesting feature: it colors squares of the board to reflect which player controls them. Red for black control, green for white control, with the color varying along a spectrum to represent weaker or stronger control (I think the formula is just Control = WhitePiecesThreateningSquare - BlackPiecesThreateningSquare, but I haven't done extensive testing). How useful that feature is I don't know because I don't know much about Chess, I'll leave it up to you esteemed gentlemen to discuss in the comments.</p>
 3456: <p>You can try the online option at <a href="https://www.apronus.com/chess/playcomputer/" rel="nofollow noreferrer">https://www.apronus.com/chess/playcomputer/</a>. You can experiment with the strength of the computer by reducing its time to think per move (in milliseconds).</p>
 884: <p>Another free chess program under Windows is <a href="http://zillionsofgames.com/demo/" rel="nofollow noreferrer">Zillions of Games</a>. Zillions is a generalized game system that includes chess as well as other games. You can easily control its intelligence. There are different settings for<br>
1. Skill level<br>
2. How variable its move choices are<br>
3. How fast the computer moves   </p>

<p>I purchased the full version of Zillions, because I wanted to play more games than those included in the free download version.</p>
 780: <p>I find that <a href="http://www.sparkchess.com/" rel="nofollow">SparkChess</a> is fast and easy to beat on its lowest level.  It has a neat flash based interface, though you'll need <a href="http://get.adobe.com/flashplayer/?no_redirect" rel="nofollow">Adobe's Flash Plugin</a> to play.</p>
 426: <p>My favourite chess engine is Numpty chess.
I am an "average"(?!) rated chess player and I have had hours of fun playing against this engine and look forward to more. It is not so highly rated like some other engines eg StockFish, Rybka, Fritz etc etc and is great for learning &amp; improving at chess. This means you do get winning chances and so you do not get so demoralised as when playing super strength engines. </p>

<p>It needs a GUI to run in &amp; is easily setup in Winboard or Arena chess gui. Chris Tatham is the engine author and a fine friend of mine (only through Numpty!).
Numpty is at <a href="https://sites.google.com/site/numptychess/" rel="nofollow">Numpty chess engine</a> and is freeware. Give it a game &amp; enjoy!</p>
 35: <p>If I'm not on my own computer, I use <a href="http://chromefree.shredderchess.com/" rel="nofollow">Shredder Chess Online</a>. You probably can't get much simpler than that. Unfortunately it has just 3 levels, but fortunately the middle level fits for me if I don't want to think too hard, and the highest level if I'm in serious mood :-)</p>
 721: <p><strong><a href="https://lucaschess.pythonanywhere.com/home" rel="nofollow noreferrer">Lucas Chess</a></strong> is great! From the description of the program:</p>
<blockquote>
<p>&quot;The computer uses different chess programs (so-called chess engines) of various strength. The user starts playing against the weakest engine at first. Initially the engine plays with limited strength but as the user wins more games the engine will be given more calculation time and its strength will improve. Eventually the engine will reach its maximum level of strength and if the user continues to win he will be passed to the next stronger engine and so forth.</p>
</blockquote>
<p>It comes with a tutor that watches over your moves.<br />
Lucas Chess ships with several chess engines from very weak to very strong (Stockfish,Rybka) and thousands of training positions (mate in 2,3,4 .. etc.)<br />
For improving your openings try <a href="http://innokuo.altervista.org/chesshero.html" rel="nofollow noreferrer"><strong>ChessHero</strong></a> or <strong><a href="http://www.chesspositiontrainer.com" rel="nofollow noreferrer">ChessPositionTrainer</a></strong></p>
 731: <p>You could try <a href="http://www.gnu.org/software/xboard/#tag-A1">WinBoard</a>. WinBoard is a GUI, that you can use with thousands of engines. Two of the most famously weak engines, that you can use it with it are:</p>

<ul>
<li><a href="http://wbec-ridderkerk.nl/html/details1/Ufim.html">Ufim</a> 0.82 The limit strength can be set as low as 700 ELO</li>
<li><a href="http://home.xmsnet.nl/vermeire/BrutusRND.rar">BrutusRND</a> This one literally makes random (legal) moves. </li>
</ul>

<p>Source: <a href="http://www.chessforums.org/chess-software-recommended-websites/9553-searching-really-bad-chess-engine.html">chessforums</a></p>

<p>When you outgrow those engines, it will be easy to switch to a stronger one. The advantage of having multiple Engines is, that they all play different and so you won't get bored as fast.</p>

<p>Additionally, you could also weaken any existing engine. Here is an <a href="http://www.horizonchess.com/FAQ/Winboard/Winboard4.html#%5BD.4.7%5D">example with Crafty</a>.</p>
 745: <p>Download <a href="http://www.babaschess.net/">Babaschess</a>, sign up for the <a href="http://www.freechess.org/">Free Internet Chess Server (FICS)</a> and start playing unrated games against real people. :)</p>
