The amount of types of IED training aids and real IED's are limited to the imagination of the builder. Limiting factors of available technology and geography drive what types of IED's will be found in certain locations. Explosives used in IED's include military ordnance, homemade explosives AKA HME, Biological, Chemical, Nuclear and Radioactive materials.
An IED can be functioned at a specific time by the use of a timer. The timer does not necessarily need to be a clock.
An IED can be functioned on demand by the use of a command detonator. The command control could be wired or wireless. Again, this depends on available technology.
Victim operated devices are also built to detonate if someone picks up the device or even opens a door.
Lastly, all of the above can be combined into one device.