I actually volunteer at a children's hospital. Volunteers play a HUGE role there. As an ER volunteer, I am pretty much assigned to help out the nurses in any appropriate way I can. I do a lot of making up of rooms, restocking supplies, etc., but I am also called upon to hold down the kids when they're getting IVs, stitches or (on occasion) having a cast put on. I also do a lot of "babysitting" when mom or dad needs to get away for a few minutes for a snack or a smoke. Other details include excorting patients up to get X-rays or to their rooms, running specimens up to the lab, etc.
Floor volunteers, I think, have a lot of patient interaction. They play in the playrooms as well as bring the fun to the bedside. I also suspect that they help out the nurses in other ways as well, but I'm not sure about the details.
One of the more coveted positions (which involves at least a year of floor experience and a detailed training) is in the intensive care nursery, where volunteers get to cuddle babies to their heart's content! (Well, within clear limits.)
In terms of "how it works"....volunteers have a lot of hoops to jump through before they can actually be a volunteer...I had two group training sessions, a one-on-one training, and a round of health clearances to pass before I was allowed to start volunteering. Limits are layed out pretty clearly (don't feed the kids unless you have permission from the doc/nurse; don't transport kids if they're hooked up to the IV; don't go in the trauma rooms when the staff is working on someone; don't violate a patient's confidentiality; etc.) You technically "answer" to the charge nurse, but for the most part I find my own things to do and am sure to ask if I'm ever in question about it's appropriateness.
At our hospital, you are required to put in one 4-hour shift per week for at least 6 months (about 100 hours). Most of the volunteers are college students who are looking to go to med school and need something nice for their resumes, but there are several "old timers" (I consider myself among them, and I'm still in my 20's) who just do it because they love it, and stick around long after they have passed the 100-hour mark. (Okay...the experience doesn't hurt on the nursing applications, either.)
On a particularly nutty day in the ER a few weeks ago, a parent asked me "are you a nurse?" When I explained that I was a volunteer, she replied, "oh, no wonder!...I was wondering why you were so cheerful!" She made me smile, but also made me realize that in these times, when nurses are pulled in SO many directions (and maybe don't get to dispense with as many smiles as they would otherwise like to), volunteers can be a great asset.