Below are some of the terms and definitions that can be found in chatting about the English Premier League. This is not a comprehensive list, but will most likely be added to gradually.
Club: A club generally refers to a specific team, however the term club is usually used as a reference to the organization as a whole. This is due to each club actually having multiple teams within the organization. The First Team is the team of regular starters that play in their main competition (Premier League). The Reserve Team is the team of basically second string players that play in a reserve team league. This team is also used to help players who are coming back from injuries. The Youth Team is the team made up of their younger players who are being developed within the club. They play in their own league and tournament competitions throughout the season. Beyond the Youth Team is the training academy that many clubs have. The training academy is usually made up of several tiers of age brackets that play in their own competitions. They are also often housed in a dormitory and take school courses in the academy itself.
Fixture: A fixture can refer to either the fixtures list or to a specific fixture. The fixtures list is the schedule for the season while a specific fixture would refer to a scheduled match.
Match: A match is the term used for a soccer game.
Supporter: A supporter is another term for a fan of a team.
Relegation/Relegated: when a team is demoted or dropped down to the next lower level league.
Promotion/Promoted: when a team is promoted to the next higher level league.
Draw: otherwise known as a tie.
Table: the statistical standings for the league.