The first type I used are called prefolds. These are most like the old school diapers that our mothers and grandmothers used (well, other than flats, but I know nothing about them, so I'm not going to talk about them). Prefolds are a lot simpler than a lot of people realize. You can either fold them around the baby and secure with a snappi or pins (I don't use sharp pointy objects around my baby...I'm all about the snappi) or you can simply trifold the diaper and lay it in the cover.
I like to use the snappi because I feel like it gives a more secure fit and keeps more poo in the diaper and off the cover...especially when you are talking about breastfed poo which is very watery.
I thought I wouldn't like prefolds because of the extra step of needing a cover. But when my boy was a newborn, they were the only cloth diapers that fit his skinny little butt, so I quickly became a fan.
They still work well and I love having a nice big stack of prefolds around. They make fantastic burp cloths. And since it's allergy season and both me and the boy have runny noses, they also make great handkerchiefs.
Questions about prefolds? Post them in the comments. I'm no expert, but I love to help people get started with cloth diapering, so I'll do my best!