In the earliest days of climbing, climbers just wanted to get to the top by whatever means worked, and there is little info about what they did. But during the 19th century, as the explorers of the Alps tried ever harder summits, it became clear that choosing, say, the east face over the southwest ridge, was the difference between success and failure. A famous example is the first ascent of the Matterhorn, which had been repeatedly and unsuccessfully attempted via the southern side, but there the strata tended to slope down and out, while the rocks of the northeast ridge (the one closest to Zermatt) tilted up - a steeper but safer route, not much harder than climbing a ladder.
As technique developed, and mountains of the world were conquered via their easiest routes, climbers began to challenge themselves by looking for and trying other routes. Once all the obvious lines had been tried, climbers looked for the less-obvious; an all-rock route threading between icefields, or perhaps a single thin crack running in a continuous straight line from base to summit. An aesthetic element came in as well; a easy but confusing route weaving back and forth across a face is less desirable than a direct route along a scenic ridge. Safe routes that yield to good technique are more desirable than routes with loose rock and awkward handholds.
Climbing routes that are unobvious and/or clever are clearly the creation of the climber(s) who came up with them, and those routes came to be known after the climbers who first ascended them. Inevitably, there were climbers so energetic that they established multiple routes on a single mountain or cliff, and it became the rule that the climber could choose the name of the route. This opened up a different outlet for creativity, and modern climbing areas will have a bewildering variety of curious and amusing names for their many routes, often following themes inspired by the area or popular culture.
As an example, one of the most famous rock climbs in the world is the "Nose Route" on El Capitan in Yosemite Valley, so named because it is at the nose of the cliff, where it protrudes the furthest into the valley. A representative selection of routes to the right of the Nose includes:
Another side effect of the large number of named routes is the need to indicate precisely where the route goes. For high mountain routes, rockfall and snowfall change the mountain so much each year that it's usually only possible to give a general idea of a route ("climb the ridge to the black tower, pass below it on the right side, and go up a snow-filled gully to the summit ridge").
Rock climbing routes are much more stable, and the wrong choice of crack can leave the climber in a dangerous position, so guidebooks will either use photographs with lines drawn on them and/or "topographic maps" ("topos" for short) showing a schematic view of the rock face along the route, using a variety of specialized symbols to indicate the direction of cracks, location of protection bolts, and so forth.