As always, Wikipedia describes it best: In computer security, Capture the Flag (CTF) is a computer security competition. CTF contests are usually designed to serve as an educational exercise to give participants experience in securing a machine, as well as conducting and reacting to the sort of attacks found in the real world. Reverse-engineering, network sniffing, protocol analysis, system administration, programming, and cryptanalysis are all skills which have been required by prior CTF contests at DEF CON.
In an attack/defense style competition, each team is given a machine (or a small network) to defend on an isolated network. Teams are scored on both their success in defending their assigned machine and on their success in attacking the other team’s machines. Depending on the nature of the particular CTF game, teams may either be attempting to take an opponent’s flag from their machine or teams may be attempting to plant their own flag on their opponent’s machine. Two of the more prominent attack/defense CTF’s are held every year at DEF CON, the largest hacker conference, and the NYU-CSAW (Cyber Security Awareness Week), the largest student cyber-security contest.
Jeopardy!-style competitions usually involve multiple categories of problems, each of which contains a variety of questions of different point values and difficulties. Teams attempt to earn the most points in the competition’s time frame (for example 24 hours), but do not directly attack each other. Rather than a race, this style of game play encourages taking time to approach challenges and prioritizes quantity of correct submissions over the timing.
LiveOverflow also gives a nice intro on his channel:
Besides attack-defense and Jeopardy style, CTF Time notes a third style:
Mixed competitions may vary possible formats. It may be something like wargame with special time for task-based elements (e.g. UCSB iCTF).
For an overview of the most used CTF’s I play(ed), you can view the CTF-sites page.