A Match is a game between two set teams of 2 or more players each, unlike a Public Game, where players join and leave at their will. Matches exist in almost every FPS game. They are often organized before the actual date, as in tournaments or ladders, others are just created when someone brings it up.
All Matches except 1on1 are in team modes, or are otherwise classified as a free-for-all.
1 vs 1
An 1 vs 1 match ("one versus one", also referred as 1-on-1) match is, as the name gives away, a match between just two players, usually on a small map.
Clan matches are matches consisting of two teams of once clan playing each other (at least 2 players on each team). These are usually started when a clan is interested in finding a match, and another team is requested.
An Inter clan is a large game usually between friends or clan players that can involve many people. This is played on a private server, unlike a pub. The purpose of these matches is to find many same-level skill players to test yourself in a group.
Inter clans are often mistakenly used to describe Intraclans.
An Intra clan is a private match consisting of only one type of clan or group. This is also called simply training or practice.