The military term is Explosive Ordnance Disposal, which includes both dealing with improvised explosive devices and explosive ordnance.
The most common methods used in bomb disposal include having a remote controlled robot pick up a suspected bomb so it can be removed to an isolated/shielded location where it can be detonated with a minimum of damage. When moving or detonating the explosive is not an option, a bomb disposal expert will attempt to disarm the device by manually manipulating its internal mechanisms in order to disable its detonation function. As a last resort, it has been theorized that one way to disable a bomb that cannot be moved or tampered with is to damage its mechanism to keep it from detonating such as shooting out its mechanisms. Of course, this also risks detonating the bomb itself.
Attempts to defuse devices manually will typically only be tried when remote methods are not feasible. As protection, the person in question usually wears a heavy specialized suit made of kevlar which can increase the person's chance of survival should the device explode, but the hands are usually exposed to allow for maximum manual dexterity in the task.