What techniques do you use to analyze your games or other people's games?
How much time do you spend on analyzing the game?
Are there any tools (electronic or manual, such as a piece of paper) you use during analysis?
Analyzing your own games is the best and fastest way to improve in my opinion.
As soon as possible after the game, write down the variations that you were thinking about during the game, especially the ones that were not played. This will be useful when you come back to the game after days, months, or even years. As you improve, it will be helpful to see what you were thinking during the game. That way, you can pinpoint how you can improve your thought process in future games.
When you start analyzing, use a chess board, but don't use an engine at first. Look for moves where you made a mistake or missed something important. If you lost material, go back and find the moment where you blundered. If you got into a bad position, try to identify the moves that led to that position and evaluate some alternatives. Put everything you come up with into your chess program/database (you are using some kind of a database, right?).
After you've gone over the game "by hand" (this should take at least half an hour for a slow game, even more time if it was a complicated position), now it's time to use a chess engine. Go through all of the moves and take note of where the engine says that you either made a poor move or missed an opportunity for a better move. Again, add variations to your database. If it's not clear why something that the engine suggests is good, investigate it further. Try out some lines and see what the engine says.
At this point, you have an awful lot of moves in front of you. Now it's time to try to take away some lessons from the game. Go back through the game again and decide what specific thing you could have done better. Did you make a very poor positional move? Did you blunder material? Did you neglect development? Did you put a piece on a bad square? Asking questions like this will help you in future games during the game.
The last thing that you can do is to ask a stronger player for help. Bring all of the information you have to the stronger player and then go through the game once more. The stronger player can give you suggestions for how to think in certain positions. Basically, a chess engine can tell you what the right move is, but a stronger player is often needed to tell you why a move is correct.
You should try to develop your own analysis style as well. If you are a strong tactical player, try to go through the game without making moves on the board, just analyze in your head. If you are good at endgames, switch colors and analyze where your opponent went astray in the endgame. Hopefully analyzing games is fun and not a chore!
I didn't mention studying the opening moves because that is less important than finding the specific points in the game where you made a mistake. After you have looked at the game as a whole, studying the opening can often help you as well. The best way to do this is to look for grandmaster games in the same opening that was played. Find out what moves are commonly played, and then go through some games of stronger players in order to find good, general plans for the opening. Try to see where you varied from these plans and determine why you didn't do what the stronger player did. This is much more important than memorizing moves (although you will probably find that you've memorized many variations just as a matter of course!).
I'd have to disagree with @andrew about analyzing ones' own games. The reason is, I'm a B-player. I make B-player moves. I perform B-player analysis. That's insufficient. Spending precious time to produce weak analysis is inefficient.
This may offend people. But consider - if you had to make a home repair and you didn't know how, would you just figure it out on the spot or would you seek an expert/education to make you smarter? If you needed to shingle your roof, would you just get up there and start hammering? Or would you refer to several how-tos and learn a little first? Maybe you'd talk to your brother-in-law who's a contractor?
Instead, I start with the engine, on a setting that I cannot beat. This is analogous to have an instructor or how-to book.
I key the game into the engine and let it analyze each move (both sides.) If the computer produces a move and the score deviates too much (for me, a half a point), I stop to determine why the computer chooses a different move. Then I work to understand how the computer's move prevents situation the caused by my move.
You have to have a certain degree of ability before you gain a lot from analysing your own games.
Certain points you can discover by yourself, but is it just a move or two, or is your whole plan bad? Often you'll not be able to answer that on your own. You can guess, but your guess isn't likely to be that much more reliable than your play in the original game. You'll learn from your own games faster if you find a stronger player to look at them with you.
If you can't do that and have to do it on your own, here's a suggested program:
Play through the game, observing it like a movie. Don't launch yourself into deep thinks. On this pass you're only looking for obvious tactical blunders, so spend only a couple of seconds looking for ways you could have lost or won material. Note them down somewhere for further analysis later. Reset the board.
Play through the game again, five moves at a time. Every five moves stop and ask yourself which side is better, if any. Mark that evaluation on the sheet. Watch, and when that evaluation changes, somebody made a mistake; go back and look through those five moves to see if you can find it. Maybe you can't, but if you can (and you've found it only if you find an alternative that maintains the evaluation) circle the move that caused the change.
Only after that should you fire up a computer to look at the game. Right now, you're checking your own conclusions against the computer. Walk the computer through the game, comparing its evaluations with your own. If they're different, spend 10 minutes or less finding out why (it's rarely productive to spend more than 10 minutes on things like this). If you can't, mark it and go on.
What you're looking for here is whether your evaluations agree with the machine, and most especially if you and the machine pick the same "turning points" in the game. You're trying to correct your thinking here, train yourself to evaluate positions better.
If, after the end of this exercise you still don't understand the computer's evaluations of some positions, set the position up and play 5-10 games against the computer, starting from that position and taking first one side and then the other from that position and see what happens. Let the computer show you what it sees, and try your ideas against it. Learn to implement the ideas and evaluations you're learning.
But seriously, the best training games to analyse are games by players a couple of classes higher than you. If you're a C/D player, Carlsen's games aren't going to do you a lot of good; that's like a a guitarist that can manage three chords trying to learn from listening to Segovia; improvement is possible, but not all that likely. A D player can much more readily understand how to play like a B player than like a Grandmaster; it's like steps in a stairway.
Game analysis is important (both your own & other people's games). I usually use some computer software-I am not great analysing only from a game scoresheet. (if so I would try & make sure I had a board setup in front of me-but the computer chess software is very handy). I have Fritz9 / Winboard / BabasChess and a variety of chess engines. I usually use these for my game analysis. It can also be very helpful to have a chess coach or other good player look over the game with you to get another opinion. Game analysis to start with is not easy but like most skills you can get better at it with practice and it can help you learn from your mistakes and successes-goodluck with your chess analysis! :-)
I would focus on the "turning points" of the game. That is, I had a good game until I made move X. If, instead, I had made move Y or Z at that point, what would have been the result?
The idea is to discover your mistakes, first the most obvious ones, then the less obvious, and try to avoid them in future games.
Engine analysis can take care of the tactical bloopers, it is usually the strategic plans and more human stuff that is hard to figure out even for strong players, and Andrew's advice should help in that regard.
About the openings however, I believe it was Edmar Mednis who mentioned that 50% of the time should be spent studying openings, and it is not just about memorizing lines but also about the associated motifs extending to the middlegame and possibly into the endgame for openings like the Spanish Exchange and Berlin. Examining opening moves with only an engine can be tricky as the playable candidate moves usually do not differ beyond one-third of a pawn (or roughly thirty centipawns) or even up to half a pawn, which means almost nothing to a human player. Cross checking the main lines with books and games between strong players is definitely more important here.
I think that the point in analysing your games is to point out in what conditions you are inclined to make a mistake (not necessarily a blunder, but also a suboptimal move).
You do not study your games to improve your chess knowledge (for that you have to study Grand Masters games), but to improve your approach to the game.
Try to determine common situations in your games, give them a name and try to find a kind of behaviour to tackle them.
Just an example:
Unexpected move: this is when your opponent made a move you did not expect and you notice that you lost confidence and you made a suicidal move to find counterplay. Then in your analysis you find out that your position was not lost at all! It was you that reacted badly in that kind of situation. Take note of it. The next time this situation arises, try to stay calm and be aware that usually a mistake is followed by a worst one in order to avoid it.
There are hundreds of situations like this that you have to find out yourself.
Create your kind of chess theory if you want to improve.
Here are my 2 cents in addition to the already well-written answers:
Start your analysis during your games instead of after. It really helps remember the critical points of the game rather than trying to recall them later (you're likely to not immediately start analyzing your game after a round finishes...). This will save you both time and effort. Here's how:
Doing this takes little time, and saves a LOT of time during post game analysis. Good luck!
The goal of analyzing a game is to understand what happened in the game, step by step. Following Kasparov's advice, you should analyze all your games, regardless of the final result (loss, draw or win). The point is that it is tempting to only analyze your losses, because you think that these are the games that reveal your weak spots. While in reality, your wins could be covering up some even more serious flaws that will continue haunting your future games.
Nowadays, analysis involves the usage of a chess engine. The ideal analysis is as follows. First, you analyze with your opponent after the game. You go through the whole game and find the critical moments and share your thoughts about these positions. Taking a few notes during this process is a very good idea. Next, you get home and enter the game in your personal database over your games on the computer. Then, you step through the game with an engine (e.g. Rybka) running to find the key moments by looking at evaluation jumps. You compare these with the moments you discussed with your opponent, to check your own understanding. You can make notes and include them in the game file. Done!
All analysis (as Gaul) is divided into three parts:
Step one has been covered fairly well by the above posters, but to reiterate:
Step two was also fairly covered but for the sake of being thorough, involves going over your variations and all the work you did during the game, looking for mistakes.
Step three is where I think some of the advice falls short.
Essentially the computer is a purely objective machine, if you use it to analyze games it may give you crazy variations which are objectively good but which have you sweating buckets. Instead, you use the computer make sure your analysis is objective (i.e, free of calculation errors, bias, etc.). When the computer analyzes the game you should only use that information to supplement your existing analyses (not replace it).
EDIT: Forgot to add a few things: