Well, it depends on the hospital, and sometimes on how old you are (for instance, they're not going to let a teen drive a cart). When I started volunteering, I was a courier. Basically, I was tasked with pushing the wheelchairs of patients who were being discharged from the hosital. I would go up to there rooms, make small talk, and whisk them away to the parking center. Being a courier was one of the main things a teen could do at that hospital. Also, other times I would deliver items (i.e., urine sample or medical equipment) across the hospital. Most of the time though, my experience was sitting down with the other volunteers and talking until somebody needed one of us. Each volunteer shift was four hours. I did mine once a week.
Another position I held was that of the mail courier. I basically delivered the hospital's mail to all of its departments. It was a good way to get a glimpse of everything, and also provided some interesting experiences.
There are different types of volunteering positions. You can usually apply to anyone you like. The website of the hospital should have a comprehensive list of all positions and sometimes what they require. If they offer an "interview" (like they did at the place i volunteer), it is an excellent opportunity to ask any questions you have.
So generally i think many people choose to do Patient Unit Volunteering, where they go around providing help for nurses and speaking to patients, making them feel comfortable. Also to my knowledge, most hospitals around where i am do not offer shadowing opportunities, so those you may have to find on your own. From there its pretty much on you to make connections and be where you want to be (but ofcourse ask the people at the volunteer office if thats ok to do). I went from being a patient unit volunteer to a position at a clinic to a clinical research position.
The hospital I volly with gave me a choice of what area to spend my time in. I chose the ED and am fairly happy with it. The time is mostly split between clerical work (helping patients register, stocking IV carts and pt rooms) and watching kids/ the crazys/ETOH crowd.
I started volunteering at this hospital a couple months ago, I handed in a form I had to fill out and went to an interview. I volunteer 2 hours a week. I choose to do patient care, which is doing things like restocking materials in the patient rooms, giving them water and bringing their food into the room, etc. Sometimes I try to talk to the patients but the conversation only lasts more than 10 seconds if the patient is not completely boring and uninterested.