Basically what happens is that random people sign up to drive for these services, then when you book a ride, the service assigns you to one of the local drivers.
It's semi-illegal in most places, but Uber has enough venture capital to not really care. (And the people who run it have kind of a shady reputation.)
I live in NYC where we've got taxis everywhere all the time pretty much, so I don't hear a lot about friends/family/coworkers using Uber. There was a bit of news last winter when they temporarily raised their prices during a snowstorm and charged people $50-100 for rides that normally cost $5-10, or so I remember reading, but I don't know anyone who experienced that firsthand.
On the other hand, last summer I visited Los Angeles, and last fall I visited Orlando, and it seemed like the people I knew in both places (especially L.A.) used it a lot. I downloaded the app in LA and used it instead of regular taxis and it worked find. As a tourist, I liked the GPS aspect to the app - instead of me trying to give bad directions or a bad explanation of where I was, the app could tell exactly where I was and relayed that information to the driver picking me up. That's a cool trick, but I see no reason a regular cab company couldn't set up something similar.
I find the term "ride share" to be weird. To me those words bring to mind the thought of a friend offering to give me a lift. When I hear "ride share" I think of someone saying "Oh, I'm heading in that direction anyhow, let me give you a ride". In my opinion, using Uber doesn't involve "sharing" - you're paying a person to pick you up where you are and drive you somewhere else. I can see why "unlicensed taxi" maybe didn't seem like the best term so someone invented "ride share" but it sure seems like a taxi service in just about every other way to me.
I haven't used it in NYC since returning home - I opened up the app a couple times here but the fare estimates were always equal or more than a yellow cab when I checked. A lot of people in a NYC have complaints about yellow taxi service here. I like a lot of the convenience of the Uber app but I'd like to see a more professional management that was more respectful of local rules and regulations.
I've heard of the service but have never used it. Frankly, I wouldn't use it even it it was available in my area. I think that there are too many liability issues on both the passenger and the driver's side in using an unregulated ride service. I doubt that the driver's liability insurance would cover a person if they were seriously injured in an accident while using such a "service".
My son has used Uber a few times. Last month, he went to Disneyworld. He and g-friend took a taxi from Orlando/Sanford airport to a hotel just outside D'world. $90 bucks. On the return trip later that week, they used Uber for the same trip. $30 bucks. Not sure why they didn't use it for the first trip. But they used it all week in and around D'world and had nothing but good things to say about the experience.