In all games except Boxing and Donkey Kong 3, you get a star for 200, 400, 600, 800, and 1000 points in each difficulty (in one game) on both Modern and Classic. This allows for a total of 20 stars from each game.
DK3's single-player mode only has one difficulty, so you can only get 10 stars from it in the manner described above. In Boxing, you get a star for every 3 victories in a single game up to a maximum of 15 wins. DK3 and Boxing have multiplayer modes, in which you can earn a star just by finishing a round with someone using the GBA Link Cable (you only need 1 game cartridge) up to 5 times per version (Modern/Classic) per game.
In total, you can earn 220 stars, but 20 of these can't be earned unless you have two GBAs. The game should explain this to you in a much simpler way eventually.