I will try and address the questions to the best of my knowledge, but please keep in mind that these are essentially 5 valid questions that should be asked separately. It's up to you (and the moderators) to keep this question open or split it into 5 questions.
What is a rec basketball team?
I believe the term rec stands for recreational, meaning "just for fun/training". Such teams are essentially much less competitive and, in my experience, usually composed of enthusiasts and old/injured players.
What is a travel team? (In basketball, and just in general.)
I have to admit I have never heard of this term, in basketball or otherwise. My guess would be that it's a team that's engaged in a league that requires the team to travel for games, for instance within the region/county/state etc.
What is the difference between varsity and junior varsity?
Wikipedia has a pretty good article about junior varsity teams, I suggest you start your reading there. Here's a excerpt:
Junior varsity (often called "JV", sometimes spelled out phonetically as Jayvee) players are the members of a team who are not the main players in a competition (such as a football, basketball, or baseball game), usually at the high school and college levels in the United States and Canada. The main players comprise the varsity team. Nearly all junior varsity players are in their freshman and sophomore years in school, though occasionally upperclassmen may play on JV teams.
I'm pretty short (5' 2") would this be a hinder in my chances getting into the team or have any significance as to what position I play in the team?
It will certainly have a negative effect, but that should not discourage you. You need to specialise, and make sure your height does not hinder your game. A great role-model is Nate Robinson who is 5' 9" and has been performing against big boys in the NBA. To make my point, here's a video of him blocking Yao Ming (7' 6"). As for positions, I suppose it's reasonable to say that you will be playing PG or SG, depending on your particular skills and on your teams squad.
What is the main thing I have to worry about over the summer, if I want to tryout for the school's team?
Well, this is a tough question, especially without knowing what you are good/bad at. I suppose you need to improve your game overall, dribbling, passing shooting.. you name it. Oh and focus on your vertical leap to negate the effect of your height. I would also "scout" the teammates, and the coach, it might give you insights as to what kind of player the team needs the most.