Jeopardy-style CTFs has a couple of questions (tasks) in range of categories. For example, Web, Forensic, Crypto, Binary or something else. Team can gain some points for every solved task. More points for more complicated tasks usually. The next task in chain can be opened only after some team solve previous task. Then the game time is over sum of points shows you a CTF winer. Famous example of such CTF is Defcon CTF quals.
Well, attack-defence is another interesting kind of competitions. Here every team has own network(or only one host) with vulnarable services. Your team has time for patching your services and developing exploits usually. So, then organizers connects participants of competition and the wargame starts! You should protect own services for defence points and hack opponents for attack points. Historically this is a first type of CTFs, everybody knows about DEF CON CTF - something like a World Cup of all other competitions.
Mixed competitions may vary possible formats. It may be something like wargame with special time for task-based elements (like UCSB iCTF).
CTF games often touch on many other aspects of information security: cryptography, stego, binary analysis, reverse engeneering, mobile security and others. Good teams generally have strong skills and experience in all these issues.
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. There are two main styles of capture the flag competitions: attack/defense and jeopardy.
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 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. One of the more prominent attack/defense CTF's is held every year at the hacker conference DEF CON.
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 solve the most number of 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.