A squire is an attendant of a knight who may act as their shield-bearer and assist them in various other ways. Traditionally, they are ranked below the knight but higher than a gentleman. There is no guarantee that they will ever become a knight themselves.
It came to also mean an English country gentleman. A related term "esquire" still exists in England to this day and is widely used in the US by attorneys (male and female!) to designate their profession.
Squires in the SCA
This is a hard topic to discuss, as squiredom can mean many things to many different people in the SCA. In my view, being a squire does NOT mean that you are any better than any other member of the society. Different knights choose squires for different reasons, and expect different things from their squires. It is something that has to be discussed when the opportunity arises. I believe squiredom is a relationship where a knight believes that he has something to offer his squire, and is mutually beneficial to both parties. At the bare minimum, this would normally mean that the knight will help the squire in their combat training, but may also aim to help the squire better understand and practise chivalric behaviour.
As I said, there are many different views with regards to squiredom and my views may totally go against those of another person.
Cecily ferch Madog's tuppence on squires:
Without disagreeing with any of the above, I would say that the most important thing in a knight/squire relationship is that both parties are clear on what their expections and requirements are from the outset and that there is ample opportunity as time passes and circumstances change for free discussion on whether the relationship is still working, and if not, what improvements could be made. A squire is not a slave and being in fealty should not mean that one must follow blindly and without the right to question or give feedback in an appropriate manner. On the other hand, becoming a squire, especially to a "hot" knight should not be viewed as some guaranteed fast-track to becoming a knight oneself, that can be readily abandoned if that doesn't happen.