A note about analyzing vs. reviewing a game: an analysis mainly consists of what is actually present in the game, not what you think should be. Pick it apart and see what choices the designer made and think about why. I figured it'd be helpful to have a template for guidance when critiquing a game. Hopefully this will also be a useful inventory of mechanics for game design itself.
Disclaimer: this is pretty long and prepare yourself for some lists. Also, appropriate use of the definition list HTML tag follows. Anyways:
Name of the game, the platform, and time played. Note that you can get a good idea of the game from playing for 30 mins or so, it doesn't have to be a 20 hour endeavor.
A paragraph summary of everything below.
How many players are supported? Does it need to be an exact number? How does this affect play? Some types of player frameworks:
Single Player - like Solitare.
Head-to-head - 1 vs. 1, Chess.
PvE - Player vs. Environment, or multiple players vs. the game. Common in MMOs like World of Warcraft.
One against Many - Single player vs. multiple (obvy).
Free-for-all - Every man for himself (1 vs. 1 vs. 1 vs. 1..). Most common for multiplayer games, from Monopoly to Modern Warfare.
Individuals Against the System - Like Blackjack, where the Dealer is playing against multiple players, but those players have no effect on each other.
Team Competition - Multiple vs. multiple, i.e. sports.
I want to start playing as many games as possible and doing brief-ish analyses of them, so here's a start:
Name: Temple Run
Time played: 25 mins
Temple run is a free (and ad free) 3D platform game. You are a man running from a herd of what appear to be gorillas and have to collect coins for bonuses along the way, all the while turning, leaping over, ducking under obstacles, and collecting coins. You can then spend your coins on powerups and new characters at a store and can, of course, buy more coins with real money if you don't have enough. There is also a comprehensive list of objectives to work through as you play.