What are the game modes, and how do you win each one?

Every match is two teams, Blue and Orange, on an arena drawn from the match's seed. What your bots need to do to win depends on the game.

Last team standing

The last team with a bot still alive wins.
If both teams still have bots at the time limit, the team with the most hull left wins.

Each team has a circle at its own end of the field, 12 m across.
The first team to get one of its bots on the ground in the enemy's circle wins. A flier has to land: hovering over the circle doesn't count.
Wiping out the enemy also wins.
If the time limit comes first, it's a draw.
Tunnelers (TUN-H) can't play infiltrate: one would dig under everything straight to the enemy's circle.

Set up like infiltrate, but each circle is half the size (6 m across).
Each team's conch starts in its own circle. Every bot has a built-in conch-cage that picks up the nearest free conch within 2 m. It holds one at a time, and never picks up its own team's conch.
The first team to get the enemy's conch into its own circle wins, whether a bot carries it in or drops it there.
Wiping out the enemy also wins.
If the time limit comes first, it's a draw.

Hide and seek is coming soon.

Ladder: when you submit your Brain code bot, you tick the games to enter it in (Enter it in). There's a separate ladder for each game, and League matches have a 3-minute time limit.
Tournaments: every tournament is played at Capture the flag.
Recent matches on the League page shows the house teams playing Capture the flag, 4 bots a side. They're good to watch to see how a game plays out.