Styles of climbing

As climbing has progressed and the climbs themselves become more difficult, both physically and mentally, so climbers have sought ways to limit the risk (and difficulty). This has resulted in a morass of terms for the style in which a climb is done. Style here does not refer to fluidity of movement or climbing in a graceful manner but to the tricks that the climber has used in his or her ascent.

In addition to this there are terms relating to the amount of equipment used and place on the rope. I’ll deal with those first.

Here the climber is alone and is reliant on no-one else should anything go wrong. There is a further subdivision:
No ropes or equipment other than that required for the medium. So rock boots for rock climbs and ice axes and crampons for ice/snow climbs. Very risky as any mistake usually results in death. This style is referred to by Americans as “free soloing”
Roped solo
The climber is alone but uses one of a variety of systems to protect him or herself.
The climber ties into the rope and moves upwards protecting himself by inserting metal wedges into the rock and attaching them to the rope using a karabiner. This system allows the rope to move freely. If the climber should fall then the protection should prevent him/her from injury.
After the leader has finished on the climb and secured himself to the rock or top of the crag, the rope is pulled up until his partner who has tied onto the other end of the rope is able to climb. As he ascends he removes the pieces of protection placed by the leader so that they may be used again.

With regards to ethics, starting from the most pure then we have:

On sight
This refers to simply walking up to the foot of the route armed with no more information than that presented in the guidebook and what can be determined from the ground. Friends and colleagues have not hinted at particular pieces of protection being required or any details of the climbing. Even better than this is an on-sight first ascent where it is not even known if the ascent is possible.
Very similar to the above except that some information on the climb has been gleaned, either from colleagues; watching someone else do the climb or having abseiled the line to check it out.
Red point and head point
These terms both refer to an ascent that has been preceeded by previous practice. In the case of red point this practice is done on the lead but falls or rests have been taken and the final ascent is a clean one, i.e. without falls or rests, from bottom to top. This is usually done on bolt protected routes as the climbing is often difficult to determine on first acquaitance.
Head pointing refers to having previously top roped a route that has little or no protection so the knowledge that you are able to do the moves in a risk-free environment allows you to determine that you can do the climb without the assurance of the rope from above.
Naturally the less practice that the climber requires, the better the style of ascent.
This is usually a precursor to a red-point ascent. It means that rests or falls have taken place and that the route was not led in one continuous push.
Not practiced much nowadays but refers to the situation where one climber attempts a route and at some point becomes so tired that they are unable to continue. They lower down to the ground or belay from a piece of protection and either try again a while later when they have recovered, or their partner ties into the rope and attempts to gain further height until they too become too tired and lower to the ground. This is repeated until the top of the pitch or crag is reached.
Top roped
The route is never led but climbed with assurance from a rope fixed from above. Thus a fall is in reality simply a sag onto the rope. Obviously safe but not considered a valid ascent. This is often done on routes that the climber considers too hard or too dangerous and has no intention of leading it. If they did then lead it, it would be a red-point or head-point ascent.

As you can see the terminology can become quite complicated though there are obvious mutually exclusive combinations. The ultimate is a naked, solo, onsight first ascent!