Wow -- you're lucky that your hospital offers doulas. There are none in my area at all.
A doula is a woman trained to take care of YOU and only YOU during labor and delivery. She is not a medical professional, per se, but is a woman who has had her own child(ren). She supports you, is an advocate for you, and counsels you as appropriate. She will allow your poor husband a break to leave the room and get a bite to eat.
Your midwife if your medical professional who manages the labor and delivery process, much like a doctor would. She typically uses more holistic methods though, and may or may not be certified (a certified nurse midwife vs. a lay midwife). Her focus is as much on the baby as it is on you. A midwife may work in conjunction with a doctor in case things get to complicated.
I'm jealous that you are being offered the option of a doula.