Jimmy Connors' backhand is sort of a third category, a "two-fisted" backhand, so called because of his fighting spirit and because of the way his opponents feel after he's used it against them. Lots of players hit two-handers, but Connors has the only one regularly described as a "two-fister". Connors' backhand is a unique shot. I love watching it, but I wouldn't recommend it; it's a relic of a different time, when players still learned the game using wooden racquets. Connors had phenomenal reflexes and hand-eye coordination, and he played at a time when most top players still used wooden racquets; that's why he succeeded with the shot, where a modern-day pro would probably fail. Arthur Ashe admired the fact that Connors drove straight through the shot, rather than brushing up on the ball like most two-handers. But today's racquets are so powerful that brushing up on the ball is the way to go. No pros today hit a backhand that resembles Connors' stroke; its limitations in the forecourt are too pronounced, and it doesn't put enough topspin on the ball. Admittedly, there are a few situations where it excels. It's great for hitting passing shots from deep in the court. It's great for blocking back hard serves. And it's great for high backhands, especially if you're in close enough to where you can actually hit down on the ball. When striking the ball above shoulder-level, Connors could take the racquet back with the head pointing almost straight up in the air, then lean in and wallop a tremendous line drive.
There is no debate when it comes to a slice backhand; the one-hander is the clear choice. Chris Evert and Jimmy Connors would hit backspun two-handers, but over the last 20 years the shot has all but disappeared. I tried it in college, and it did offer one nice advantage: it made it very easy to put sidespin, as well as underspin, on the ball. When I'd hit a two-handed slice, the racquet would be descending and starting to come across my body to the right when I made contact with the ball. The resulting shot would often fool my opponent; even if they were accustomed to backspin, they weren't used to having the ball jump to their right after it hit the court. Being able to impart the extra spin is nice, but in other ways the two-handed slice is too limiting. It requires the same fancy footwork as the two-handed topspin backhand, but it isn't the mighty offensive weapon that the topspin two-hander is. People accept the limited reach of the topspin two-hander because of what they get in return: a powerful, heavily-spun backhand. With the two-handed slice, you get the limited reach of a two-handed shot without the big payoff. Often, the reason you hit a slice backhand is because you're stretched too wide to go for a topspin shot, so it's best to get used to hitting it one-handed, which gives you extra reach and allows you to conserve energy when you're just trying to stay in a point. A one-handed slice backhand is a natural, almost effortless shot to pull off. For most recreational players, it's their bread-and-butter backhand.
If you aren't willing to work hard to get into good position, then the two-handed backhand definitely isn't the stroke for you. It isn't as adaptable to balls of different heights as a one-hander, forcing you to take small steps which will put the ball in the proper hitting zone. (The good news is that, since you strike a two-hander further back than a one-hander, you have an extra split-second to take those little steps.) Obviously, the limited reach of the two-hander is a drawback. Also, if your opponent hits the ball hard and right at your body, it's all but impossible to hit an effective two-handed reply. I remember seeing Boris Becker hitting powerful one-handed topspin backhands where the ball was a couple feet in front of the middle of his chest. Of course, that's not where he preferred to hit them, but his one-handed backhand made those shots possible under duress. A player with a two-handed backhand would be forced to slice the return in the same situation.
Pete Sampras used a two-hander when he was young; from what I've heard, it was his best shot. Then, as a teenager, he switched to a one-hander, because he wanted to win Wimbledon someday. Wimbledon is played on traditional grass courts, a very fast surface; the ball tends to squirt through the grass fast and low. To win Wimbledon, you need to be able to adjust on the fly to unpredictable bounces. The best strategy is to get in to the net and not let the ball bounce at all. Bjorn Borg had a two-handed backhand, and he won Wimbledon five times in a row. But in general, Wimbledon favors one-handed players, at least on the men's side. Players with one-handed backhands generally volley better on that side than two-handers. The great volleyers of recent times -- McEnroe and Edberg, and to a slightly lesser degree Sampras and Becker -- all had one-handed backhands. The exclusive use of the one-hander seems to have given them better feel on the backhand volley.
The one-hander is more versatile than the two-hander. I think it's hard to argue with that. The reason so many top players use a two-handed backhand is that it is extremely good at certain things. It makes it easy to put topspin on the ball, and to control the amount of topspin; you can hit blistering drives or big looping moonballs almost at will. You can disguise your shot well, and change the ball's direction with ease. And if an opponent blasts a hard shot into a two-hander's strike zone, a two-hander can usually blast it back just as hard. When facing an opponent who has lots of pace on their shots but who can't aim them well enough to handcuff you or stretch you out wide, you should do well with a two-handed backhand. It's very hard to overwhelm a two-handed backhand with pace alone.
One of my favorite aspects of the two-hander is that it makes it easy to hit topspin lobs, which are tricky to pull off with a one-hander. Mats Wilander, a two-hander who won 7 Grand Slam titles, had the best backhand lob I ever saw. He took the racquet back just as if he were going to hit a passing shot, then suddenly swiped up on the ball and flipped that racquet head around as if it were an arm of a windmill. He could hit winners all day long that way against the best net players in the world. I remember seeing him in the semifinals of the '88 Lipton against Yannick Noah. Noah was 6'4" and could jump like Michael Jordan, but Wilander's lobs were so high and had so much topspin that Noah didn't have a chance. It was a remarkable display.
All in all, I think that the one-hander and the two-hander have equally-balanced pros and cons, provided you're in good enough shape to do the extra work that the two-hander demands. On the other hand, if you have relatively weak forearms, you may be forced to adopt a two-hander in order to hit aggressively from that side. Most of the top women have two-handers, not because they aren't strong enough to hit one-handers, but because when they were young they needed the extra stability of the two-hander. Justine Henin-Hardenne hits a beautiful one-handed topspin backhand that John McEnroe says is his favorite shot to watch these days. Martina Navratilova had a one-hander (though she usually sliced it), and she won 9 Wimbledon singles titles. But almost all of the top women hit their topspin backhand with two hands, even players like Jennifer Capriati and Serena Williams, who are clearly strong enough to hit a one-hander. So, if you're a woman, I would recommend you try the two-hander first. If you're a man, I think it's 50/50.
A note on racquets: If you do choose a two-hander, try to get a racquet with a long-enough grip to let you grab hold with both hands. Some racquets have short grips that aren't really compatible with a two-hander. With a two-handed grip, most of your left hand should be holding the grip, not the frame.