Forget all you have seen in the films. Particularly those involving macho men in tank tops, and teams of climbers ferrying high explosive up Himalayan peaks.
Rock climbing involves coordinated and controlled movement using the natural features on the surface of the rock. At times the climber will have to work out which holds to use, the sequence to use them in and then perform that series of moves. As kids you would probably have gone through that process every time you climbed a wall, building, tree etc without a great deal of thought - particularly to the consequences.
Being older and wiser we all know that falling any height onto a hard surface hurts hence the 'mature' climber also has to deal with the stress associated to this hard wired psychological response to falling.
'Trad' (traditional) Climbing
I mentioned managing risk being an integral part of the climbing experience. This is the key to why climbers generally do not end up as organic jam at the base of climbs, even if they fall off.
A key element in 'trad' rock climbing in the UK is protecting yourself. This involves using some very specialist bits of kit to place in naturally occurring cracks and pockets on the rock. You then clip your rope to these via an extender(1). This complete set up of protection and extender is then referred to as a runner.
Your eagle-eyed partner at the bottom of the climb uses a device called a belay plate(2) to pay out the rope as you ascend. If you blow it at the sharp end, you take wing, and if you have done a good job of protecting the route, and your partner if good at belaying, you only fall twice the distance you have climbed past your last runner. This whole process of being on the sharp end of the rope (minus the falling bit) is called leading. That last part - the obeying the laws of gravity bit - has a wide range of terms from lobbing off (for a short fall) through to screamer (self explanatory I think).
Of all the climbing disciplines 'trad' climbing is the more complex activity to become involved in primarily due to the equipment and safety systems involved. It is also the most prevalent style of climbing outdoors in the UK.
Although it will take time, developing a good solid background in 'trad' climbing will open all sorts of mountaineering doors to you - especially to climbing on snow and ice, the alps etc.
Sport climbing is radically different from 'trad' in that all the protection is already 'in situ'(3), commonly bolts. It is very prevalent on the Continent particularly in France and Spain, although there are an increasing number of 'sport' crags in the UK.
All you have to do is clip your extender to the bolt, clip the rope through the extender and on you go secure in the knowledge that your eagle-eyed belayer is still doing a good job. This style of climbing has the advantage in that it is very quick and easy, technically, to pick up. Although bare in mind that there are still key skills that require specific training and practice.
Bouldering allows the climber to concentrate on the pure difficulty of short sequences of very dynamic and powerful climbing, often only a few meters of the ground. It takes place on boulders or small outcrops, without the clutter of ropes and hardware, which means it is just you and the rock.
Working out how to use the sequence of holds effectively provides a mental aspect to what is a very physical activity. In addition to being a very absorbing pursuit in its own right, it is also a good way to hone your rock climbing skills without the distraction of worrying about how good the protection is, or worrying about taking a fall.
Just a cautionary word - although bouldering is seen as being 'low risk' it is still possible to hurt yourself through landing badly on uneven surfaces when you jump/fall off.