A greater list of actual positions can be found here.

What is the minimum number of people that you need to shoot a film? One. You.

However, having more people can help. But beware: "The more the merrier" is a falsehood. I personally like working with small crews of 3 people (maximum around 7). That way everyone is involved and important. Roles are easier to keep in check. And it's easier to coordinate three people's schedules than 35.

Who is best to get? Those that you can.

This may sound trite, but there is truth here. I would take my friends who didn't study how to make movies over a film major most any day. This is because my friends will make time to help out and will be there. Not only that, they will be happy to help in every and any way they can because they love me. On the other hand, Mr. Film Major will not be thrilled to be on my set, and if a better film comes up, he'll gladly leave mine. Nor will he want to listen to me when he thinks his idea is better. In short, if it's your film, you don't want to fight another film maker's ego on set. Get people who know nothing and love to learn. That's what my team looks like: I'm the only one who has studied film. Besides, the best training you can get for making movies is gained by just doing it.

Find people with talents that match up with the position you want them to fill. For example, I had a friend of mine who loved taking digital pictures be my cameraman. And he was great! I showed him the buttons that I knew about on my camera (um, "Record"... er... "Focus"), and he figured out the rest. I got great footage. And he has never even taken a photography class.

Here are some things you want in those who make up your crew:
Someone who is your friend, smart, creative, has a good attitude, is hard working, loves to learn, and wants to help.