A midwife takes the place of your OB/GYN for pregnancy and birth related stuff. Unless they are ALSO a doctor, midwives do not have surgery privileges so shit starts going down-hill they will be replaced by a doctor at the hospital. Many midwives are technically nurses as far as education/registration is concerned, although there are midwives that are neither nurses nor doctors.
Midwives are great if you don't have any complications or any sort of high-risk pregnancy and are looking to have a natural, vaginal birth. In general they are far less likely to push excessive medication and have far lower C-section rates than OB/GYNs. Also they generally will have a much lower patient load (some will only care for one pregnant lady at a time) so you tend to get more individualized care and your midwife will more than likely actually be the person to delivery your baby, unlike your OB/GYN who, unless you are lucky enough to get them to special you, may not even be there for your birth.
Oh, midwives are generally a hell of a lot cheaper than a doctor too if that is an issue for someone at all.