Basically, it all depends on the cleverness of the bomb maker.
The most elementary bomb (ignoring firecracker-cartoon string fuses) is an electrical blasting cap stuck in a less volatile explosive like C4. Safe to move, safe to carry. Hook the wires to something that runs enough current through the blasting cap, and run the current, and “boom”. The current may be a electronic timer, a mechanical timer closing switch, a cell phone where the ringer triggers, or whatever. In this case, cut any wire from the battery, or one of the two wires to the blasting cap, and no current triggers the cap. Pull the cap out of the explosive, and a small explosion of the cap is all that happens.
Then, if you think someone will find and disarm the bomb, what do you do? You can put a dummy blasting cap with a spring switch in the bomb - pull it out, and the switch springs closed.
Maybe you put battery, cap, and a sensor inside the explosive lump. Someone cuts the wire, the circuit buried in the plastic explosive detects this, triggers the blasting cap.
Maybe you have a movement sensor - a mercury switch where if the bomb is jiggled or tilted, the switch closes, boom. (Unless you’re a Hollywood script writer for “Speed”, where you put this movement sensor in a bus doing 60mph on well-paved LA streets.) maybe the spring switch is something the bomb sits on, if it is lifted it sets it off. Classic “Bomb on the plane” movies had a pressure sensor, once the plane was over 5,000 feet, the bomb would arm and then go off when the plane descended. (ah, Hollywood) Maybe it’s a light sensor, or a magnetic switch on the case the bomb is in, or… One trick with these is to build a clay dam around the bomb, pour in liquid nitrogen to (a) freeze mercury switches, and (b) kill the battery. So add a temperature sensor.
You can dream of a hundred ways to booby-trap your bomb. The question is, why bother. Is the idea to have it go off randomly before it is detected? or to booby trap it and have the bomb squad’s robot blown to pieces (or not). The danger with any “trick” bomb is having it go off when you arm it, because the spring switch or sensor or whatever was not set right. (I vaguely recall news items about this, was it Ireland or the middle east?)
The NJ bomber apparently could not get his hands on blasting caps, it seems; instead used a broken Christmas light bulb set with some much more volatile explosive mixed in to set off the rest of the explosive. Light bulb filament sets off explosive.
The thing with any of these, you want to test them to be sure you don’t set yourself off prematurely.
(Then there’s the case of the president of Pakistan many years ago. A minute after his convoy drove off the bridge, the bomb exploded. The convoy had a cell-phone jammer as standard equipment, so the cell call to detonate the bomb only went through once the convoy was clear.)