This is something that can be either taken seriously, or simply a part of the game that SCA is. For some, their consort could be their boyfriend/girlfriend, husband/wife, for others it's simply a friend. It is generally a fighter and his lady/her man, although if they are both fighters they can be consort to each other. Either way, to formalise the friendship, favours are exchanged, and should be worn whenever the fighter participates in combat for the consort.
Because being a consort means different things to different people, before coming a consort a few things should be discussed such as how long you plan to be consorts for (one tourney, until one of you leaves the SCA, whatever), what the expectations are (water bearing, being present at all tournaments). If the person you are to be consort to is in a relationship, does the other person know? (some people can become awkward about such things).
During court, if your consort is not present, you must be his/her representative.
When it comes to something like a Crown tournament or Coronet tournament, there are some other things that should be considered. To enter a Crown Tournament, both the fighter and his/her consort must be SCA members to be eligible. They must also understand and meet the Kingdom Law and Corpora requirements for participating in the tourney, and be willing to take on the responsibilities of becoming king/queen or prince/princess.
An informative site about this topic is http://www.atenveldt.com/articles_forms/a_consort.htm