CSAW CTF, a Jeopardy CTF, has some of the best collegiate hackers in the nation
Jeopardy CTFs are the most common kind of CTF.
They revolve around a set of challenges which are provided by competition organizers to competitors.
Competitors form teams and then work on the challenges together.
Each challenge is designed so that when the competitor solves it, a small piece of text or "flag" is revealed. The flag is then submitted to a website or scoring engine in exchange for points. The amount of points rewarded is typically relative to the perceived difficulty of the challenge.
Competitors usually receive about 72 hours (typically the course of a weekend) to solve as many challenges as possible.
Attack & Defense CTFs
DEFCON CTF Finals, an Attack & Defense CTF, is widely considered the world cup of hacking
Attack & Defense CTFs are a less common kind of CTF with more moving parts. They're rarely done for the general public because of their complexity.
In an A&D CTF, teams are each given the same set of vulnerable server software. Teams are to setup & audit this software before the competition. At the start of the competition, teams will connect their servers to an isolated network to join the CTF.
Within this network, teams will launch attacks against each others servers hoping to exploit the vulnerabilities they've found. Likewise, teams will need to properly patch their software so that it is protected against these exploits and functions normally.
Teams receive points for extracting flags, properly defending their flags, and keeping their servers operating normally.