Do skylights leak? No. That was easy! Ok, I have to add some caveats- if they are of an older design-maybe. If they are installed wrong- certainly. If they are placed on the wrong part of a roof, like in a valley (the concave intersection of two roof planes)- maybe. So, the bottom line is that a modern skylight properly installed in the correct part of a roof is not going to leak.
Does that mean skylights are a good idea? Maybe. I use them a lot, because I love daylight and skylights are usually the easiest and least expensive way of getting it, compared to dormers or other structures that project out from the roof to hold a window. But they do have their drawbacks. First, they are not energy-efficient. We take pains to put lots of insulation in our ceilings and/or roofs, then punch a big hole in that insulation. A skylight is going to transfer heat to the outside about fifteen times more than the well-insulated section of roof that it’s in.
The black hole: At night, skylights are not that pleasant, unless a full moon happens to be shining in. They tend to read like a black hole in the ceiling, or like a strangely angled mirror. This effect can be mitigated with a shade (which I recommend), but unless you get an electric shade they are often hard to reach and operate. Another problem is sound, and specifically the sound of rain. If you live in a noisy area, a skylight is going to let some of that noise in. Worse, they can be very noisy in the rain. For that reason I avoid putting them in bedrooms or TV rooms.
A laylight is a great trick that solves many of the issues mentioned above. It is generally used when the skylight is on an angled roof plane and the ceiling of the room is flat (not cathedral). Typically, a flat ceiling has a light shaft up to the skylight. This shaft is somewhat costly to build and finish, and can be a little odd-looking. A laylight is a translucent panel that lies horizontally at the bottom of the shaft. Light passes through it, but the translucense limits the view up the shaft. A laylight can be very ornate of quite simple, depending on whether you want it to be a design feature or simply add light to the room.