There are 3 types of climbing rope out there: single, half and twin.
Singles can be used on their own, and are fine for sport, climbing walls and shortish trad routes. They vary in thickness from about 9.5 to 11mm.
Half ropes (sometimes called doubles because you need two) are lighter and thinner (typically 8-9mm) and are used for longer trad routes where the pitches don’t necessarily follow straight lines. You’ll need two as one rope is clipped into each piece of gear in order to give two (hopefully) straightish lines and so reduce drag. Halves also allow full rope length retrievable abseils by tying them together and abbing on both strands.
Twin climbing ropes are lighter and thinner still (7-8mm), you need two and both ropes must be clipped into every piece of gear. They are used pretty much exclusively for ice climbing in order to give an extra margin of safety if one climbing rope gets cut. They also allow full length abseils by tying both ropes together and abbing on both strands.
Important factors to consider for all climbing rope include impact force (the amount of force transmitted to the climber and gear when a fall is held – generally lower = better) and weight and durability (you’ll usually find a trade off between the two). Dry treatment is useful if you climb on sea cliffs, ice, bad weather or the mountains as wet or frozen ropes are significantly weaker, heavier and a pain to handle.