There are two basic ways of ascending a rope.
Prusik Loops
Every climber, whether they are climbing on a tiny outcrop or giant alpine face, should carry two prusik loops. These should be a loop formed from 1.5 metres of 5 or 6mm cord, tied with a double fisherman's knot. These should be clipped or 'larks footed' to the rear loop on your harness at all times, as they are a vital part of your self-rescue equipment.
Prusik loop tied with double fisherman's knot
If you find yourself hanging in space, or standing at the bottom of a rope (or ropes) that have to be climbed, remove both prusiks and wrap them around the rope(s) to form a prusik knot. One knot will be attached directly to your harness, while the other forms a foot loop. Extend the foot loop by clipping 60cm into it (or carry a 2.5 metre length of cord instead). Place the foot loop below the waist loop, stand down on the foot loop and push it up. Once weighted, the prusik should lock down tight around the rope(s) and allow the foot loop to be brought up and the action repeated.
Mechanical
The prusik loop is the perfect tool for the unplanned bit of rope climbing - it's cheap, lightweight and low tech. However, it's also very slow, hard work and often scary. Mechanical ascenders on the other hand are heavy, expensive and slightly more complicated to use. The difference is that they are dedicated rope climbing tools and make ascending even long, free-hanging ropes, a relative doddle.
Mechanical ascenders make ascending even long, free-hanging ropes, a relative doddle
There are two categories of mechanical ascenders
Firstly there is the modern, lightweight, mechanical prusik or mini ascender. These include the Petzl Tibloc and Wild Country RopeMan models. These are small, light and relatively cheap and are very effective emergency rope climbing tools. They are used in the same way as the conventional prusik system described above and are useful for climbers who have a greater likelihood of having to use a prusik.
A dedicated mechanical ascender is designed for difficult or long rope climbing pitches, like you may find in caving circles (SRT), on fixed rope expeditions or big walls. These devices, like the classic Petzl handled ascender and Croll, are very easy to use and can easily be clipped and unclipped from the rope. Most models are fitted with a handle to aid comfort, and cams that will bite down hard on muddy ropes.
The system
The most common rope climbing system is to wear an unhandled ascender (Croll) clipped directly to your harness and held in place below your rib cage with a chest harness. A handled ascender is clipped to the waist via a short cowstail or sling, and tied so that when weighted the arm holding it is still very slightly bent. A foot loop is then clipped into this ascender. When both ascenders are clipped to the rope (Croll below handled) the climber puts both feet into the foot loop and stands up. At this point the Croll slides up and locks down while he lifts his legs. By pumping away with the legs the climber can ascend a rope very quickly and with minimal effort.
Even if you have mechanical ascenders still carry at least one prusik loop for abseil protection
Safety note
Even if you have mechanical ascenders carry a prusik loop for abseil back-up and don't forget when you clip your ascender to a rope, always tie into the lead rope every 10m just in case a prusik fails.