Although you can wear any shorts for cycling, if you intend to spend a bit longer in the saddle or cover any distance - it is worth treating yourself to a proper pair of bicycle shorts!
Most bicycle shorts are padded to help protect your nether regions from getting saddle sore.
The two main types are;
Skintight Lycra Bicycle Shorts
Baggy Bicycle Shorts
Lycra road bike shorts are more efficient as they don't flap about in the wind while you're riding your bike - perfect for road riding or covering long distances.
There are no pockets in them, so they are best worn in combination with a cycling jersey - bike jerseys have their pockets at the back - out of the way when bent over the handlebars...
For mountain biking most riders wear baggy padded bicycle shorts (XC MTB racers still wear Lycra).
These are usually made of a more durable rip-stop material than road shorts. The more you pay, the more features and details you get. Pockets and vents increase in number as the price goes up.
On cheaper baggy shorts the padding will be stitched-in. For a bit more money you will get a removeable padded inner short. This allows you to wash the liner separately - handy for a cycling holiday or a weekend away.