Boardgamegeek ( http://boardgamegeek.com ) has a play reporting feature which can include who played, what position each played, their score, and additional notes.
Play notes can be publicly accessed.
(It can be a bit of a pain, tho, to find someone else's plays.)
It does exactly what you're asking for.
If the same person reports every time, it's readily viable to find it.
Further, one can create a private geeklist which lists the games, instead of using the play reports feature.
boardgames.stackexchange.com/users/1047/aramis
NemeStats ( http://nemestats.com ) is a new site that is purpose-built to be able to record the results of any game to get interesting statistics like win/loss percentage, list of players
you beat the most,
players you lose to the most,
games
you play the most, your average rank in games, etc.
It is mostly geared toward players that play in a stable gaming group but could also be used to record just your own plays against anonymous opponents if you wanted.
**Update as of 2016-02-26: We continue to enhance the site and also have the ability to import your geek list directly into NemeStats to lower the barrier to start creating games.
**Update as of 2016-07-20: We now have integration with the iOS app Board Game Stats -- providing the best web + phone app experience out there!
boardgames.stackexchange.com/users/9887/jakejgordon
It seems that Rankade.com suits your needs.
My friends and I use it to keep score and rankings from over 100 games, and with over 600 matches.
You can see the public page of our group here .
It's free, with apps for mobile (both iOS and Android) and webapp (which I like more, as it has more charts and features than the app).
I also have different groups for sports or other games, not only boardgames.
For example, I think it's nearly perfect for table football and table tennis.
boardgames.stackexchange.com/users/12893/Aubrey
I realize I'm late to the party but check out shpeely.com .
It allows to track all your games and creates statistics for each game and player.
Also it has a scoring system that allows you to track your performance across games.
boardgames.stackexchange.com/users/15552/user15552
This is an iOS app (your question mentions not only a website, but a 'system') which is awesome for all boardgame tracking.
It has great interface, lots of stats, integration with Boardgamegeek and Nemestats, and much more: Board Game Stats .
(I'm just a fan of it)
boardgames.stackexchange.com/users/5583/Dennis Golomazov
Here is an App that keeps score for you!
Keep Score - Score Keeper
boardgames.stackexchange.com/users/3215/meltdownmonk
I use the score geek app
and it's pretty good.
It syncs your plays with board game geek to and has pretty much all of their library available to score.
It's not free though.
Scoregeek website
boardgames.stackexchange.com/users/3431/Jean-Sébastien
My friends and I are pretty competitive.  We play a lot of various games with various scoring systems used by each (i.e. 7-wonders, Catan, Dominion, etc...).  Does anybody know of any website or system that can be used for keeping track of who won which game and by how much?  Doesn't need to be applicable to all games, but would be nice to keep a record.
Are there any websites for keeping score of various boardgames?
boardgames.stackexchange.com
<scoring>
boardgames.stackexchange.com/questions/12720
boardgames.stackexchange.com/users/1866/Andy S
Boardgamegeek has a play reporting feature. Nemestats was purpose-built to record the results of games and get interesting statistics. Boardgamegeek record what position each played, their score, and notes. NemeStats  is a new site that records the results of any game and offers interesting statistics like win/loss percentage, list of players. Then there is Rankade.com, shpeely.com and the score geek app.
Boardgamegeek has a play reporting feature which can include who played, what position each played, their score, and additional notes. NemeStats  is a new site that is purpose-built to be able to record the results of any game to get interesting statistics like win/loss percentage, list of players.  Then there is Rankade.com, shpeely.com and the score geek app.