A brief explanation of types of competitions commonly run by the EBU
Match-pointed pairs are just like the most common club games people are used to, where you play with a partner and your score on each board is based on how many of the other pairs you beat. Any pairs event can be assumed to be scored like this unless otherwise specified.
Teams of four has each pair teamed up with another pair and they play the same boards in opposite directions to their team-mates against the same opposing team. They score up by simply adding the scores of the two pairs for each board. Usually this is then converted to IMPs and sometimes the overall score is then converted to Victory Points. Some teams events are scored by Point-a-Board, but this is not common.
Although it is possible to have teams of eight or even twelve, if an event is just said to be teams it should be assumed to be teams of four. This is further complicated by the fact that some long teams of four events allow up to six players in a team, though only four of them play at a time!
Swiss Pairs events as played in the EBU have pairs playing long matches (6-8 boards) against each opponent after which they are assigned new opponents based on their score to date. This means you usually play against people of a similar standard and it increases the likelihood of you winning a match. Although scoring is match-pointed as above, the match score is converted to Victory Points (0-20).
Swiss Teams follow the same principle above of being assigned opponents based on your current score, but in the case of teams the scoring is IMPs converted to Victory Points.
Championship Pairs usually have a qualifying session or two of match-pointed pairs after which the top pairs qualify to play against each other in a final, while the remainder of the field, and any late joiners to the event, play a consolation event which is usually Swiss Pairs.
Multiple Teams has a movement like pairs and is usually short matches, but you play the same boards as your team-mates, in opposite directions, against the same opponents, so that you can score up as usual in teams.
A fuller guide to these sort of bridge terms can by found on the Somerset website.