One system used every day in Seychelles when digging trenches is to cover the object or rock with a mesh of old car tyres which will dampen the blast but allow escape of the gases. The explosion is thus under the control of the operators and similar equipment is used in the UK but is more sophisticated in its design.
Speaks for itself. If an explosion can be controlled, then the area can be evacuated and property protected. Objects which may cause more damage if sent flying by an explosion, such an cars, can be moved and it's even possible to protect windows with sandbags. A bomb is only detonated this way if it can't be defused safely as last resort. Bomb disposal workers are skilled and very brave people and I trust their judgement. Give me a controlled explosion any day, Matey.
Most controlled explosions don't detonate the bomb they blow it appart with a small charge so as to not set off the big one!
A bomb works by having a detonator,a small device which explodes relatively easily, explode in contact with the main explosive charge (which itself is much harder to get to explode). In a "controlled explosion" the object is to disrupt the structure of the bomb so that, for example, the detonator is separated from the main explosive charge, or the fuse (which ignites the detonator) may be separated. This can even be done with powerful blasts of compressed air in some cases. The principle is to input enough energy to disrupt the device but not enough to set off the main charge. You have to know your bomb!
The most common method of controlled explosion, used on the UK Army's 'Wheelbarrow' - a remote-controlled robot - is a shotgun shell, filled with water rather than the more ususal lead shot. As previous respondents have stated, the idea is to break the components of the bomb (timer, trigger, explosive) apart faster than the detonator can trigger the explosive material.