* Shorts are made from flexible lycra material sewn together from multiple panels, sometimes with different colors. The seat and front is padded with leather (hard to find and not useful) or synthetic chamois. Sometimes the padding contains gel cushions, but this is a bad idea because they do not let sweat evaporate to the outside, and they do not even improve comfort noticeably. Shorts come in male and female versions that are cut differently, and have different padding. Bicycle shorts are not used with underwear because that would defeat the advantages.
The advantages of bicycle shorts are that they are flexible, padded, and seamless. Regular pants may constrict your thighs (which expand when you ride), and they will chafe the inside of your thighs where they rub against the saddle, especially if they are loose-fitting because they bunch up. I never ride anywhere without bicycle shorts. If you are concerned about looking geeky, there are versions with an outer part that looks like regular shorts, with invisible inner lycra shorts.
Make sure that the waist and leg hole size fits. Shorts must fit tightly but may not constrict the waist or the thighs (sometimes leg holes are far too small). Waist straps are useful but not necessary. The legs should have rubber threads sewn in at the leg ends. It usually looks like multiple thin white rings stitched into the inside. Do not buy pants that just have a single rubber ring sewn into a fold, they slip.
* Long pants come with and without padding. The ones with padding are like shorts with long legs, and the ones without are worn over regular shorts. I prefer the latter because I don't need as many of them, because I can take them off when it gets warmer, and because I have found the padded variant to slip easily. Pants must be close fitting, and must be long enough to pull up at least over your navel (you don't do that normally but if you can't they will slip). I prefer pants with a cord at the waist that keeps them up better. (I have found such cords to be unnecessary in shorts.)
Long pants have the same advantages over street clothing as shorts, with an additional one: in wet weather street clothing becomes heavy, inflexible, and clings to the skin. Bicycle pants remain flexible, and although they get wet they don't soak up water, and the reduced evaporation and the fact that there is no air between skin and pants make them feel much warmer despite the fact that they are made from much thinner material.
Long pants come in various thicknesses, from the thin lycra material used in shorts to thicker lycra with a fuzzy inside, and as fleece. I have them all: the thin ones for warm weather down to about 10 degrees C, the thick and fuzzy ones for temperatures above freezing, and the thick fleece ones for anything lower than that. (Fleece does not work well in rain.) Some of them have zippers in the lowest part that makes them easier to get over the feet (or even shoes); the material is flexible but pulling too strongly will tear the seams.
* Rain pants go over long pants and are made from waterproof material. They are very uncomfortable on skin because they are basically sturdy plastic bags. All the ones I have tried begin to leak after a few minutes, and they get in the way when riding. I recommend against them. The only marginal advantage is that the spray from rear wheels without mudguards does not soak the back of your pants but I have found it to be almost as effective to stuff a plastic bag between shorts and long pants.
See also an article on "Shirts"
