1 - your "normal" taxi, the kind most Westerners think of when they hear the word taxi, a car that is clearly labeled as a taxi and you can either flag down or call a service and have a taxi meet you wherever you are. These are, of course, the most expensive, and you can't really bargain the price--there's a meter.
2 - a taxi that belongs to a service, and thus can be called like the normal taxi, but it's usually just a regular, unmarked car, usually one that's rather old. Some of these have meters too, but you can sometimes bargain too. (Asking for a price at the beginning of the ride and then sticking with it.)
3 - any car that sees your outstretched hand and decides to stop for you. Anyone can be a taxi here. Some people are what I call freelance taxi drivers, they don't have a real job, they spend their day in well-traversed areas such as airports and get customers there; they might drive around the city looking for people with outstretched hands. A freelance taxi driver may give you his number, so you can call him. Others, you just hail on the street. It's best to bargain a price before you get in. They are often male and rather fast. It's quite similar to the Western version of hitch-hiking, except that it's so common here, and people think I'm weird when I say it worries me.
I often take the last kind when I take a taxi, and I have finally gotten comfortable with it. This past Friday, a friend and I planned to go to a violin concert, after which I would go to a St. Patrick's Day event at the US Embassy, and since I planned on drinking, I decided to take a taxi. We took a taxi to the Organ Hall, only to find out that the venue was not the Organ Hall (where most of the classical music concerts are) but rather at the new building called Shabyt, a building that was being built my first year here, and is large and blue and circular and I think is sometimes referred to as the "dog-bowl" building (a lot of buildings have weird nicknames).
Shabyt is quite a ways away from the Organ Hall, but we had arrived early and had almost 30 minutes. We got in the first car that stopped for us.
My friend asked the driver if he could go quickly, and the driver gave some explanation as to the ice being slippery. If a taxi driver has an excuse for not going fast, that's a bad sign; he was the first taxi driver I've ever encountered who preferred to go slow.
He went slow, really slow, and other cars passed him. I could drive faster than him (and I'm an incredibly slow and cautious driver). Also, he had another passenger, who sat quietly for most of the ride.
We did not drive in what I would think is the most direct route, but I figured, it was rush hour, maybe he knew a less traffic-y route. But he kept driving in the wrong direction, so finally I had my friend say something. He said he was going in the most direct route, I did not know what I was talking about. I suggested we hop out, and I wish we had.
He was going to drop off the other guy first, but after we protested so much that we were going the wrong way and that we had to be there by 6, the other guy said he could drop us off first, and helped give the guy directions.
The trip ended up taking 50 minutes. It should have taken 20 - 30 minutes, depending on traffic.
During the trip, the guy talked to my friend most the time. She was sitting in the front, and her Russian is better than mine. I understood about half of what he was saying, and she translated some more for me, and well, he was rather rude and inappropriate. At one point he asked her how much money she makes, and when she said she didn't want to say because that's personal information, he kept at it, asking again and again and insisting that she tell him.
At another point he asked if she had children and told her that she should have five children because that's best, but she argued with him, why should she have five children? Who said that's a good number?
He asked for her name, and she said she didn't want to give it, and he said why, it's the young men you should be scared of, but he's old, like a grandfather.
In the end, we were quite glad when we made it to the Shabyt, twenty minutes late for our concert.
We were just so happy to get out of that car!
At least we got a story out of our adventure! And the concert had started late (as they frequently do) and had not sold out of tickets (as they rarely do) so we managed to enjoy a nice concert in a new venue.