Is there a specific cause of a "hydraulic problem" or could it be a lot of things?
 

I read, or hear ATC audio on YouTube, of pilots returning to the airport due to a hydraulic problem, loss of one of the hydraulic systems, etc. Is there a typical cause of this failure or could it be a lot of things?

"Hydraulic problems" can be lots of different issues; it is the kind of wording one puts in a press release because the writer does not know any detail.

"Loss of a hydraulic system" means that the controls operated by that system are unavailable for the pilots. Nearly always a lack of pressure due to losing pump capacity or losing hydraulic fluid. (Fluid leaks happen)

Yes can be a myriad of things from a sticky valve to a pump to simply an electrical supply to the system or an indication light, don't worry as aircraft have redundant systems as back ups so if you lose one the other one or two take over. But obviously you ideally want all of the systems working in case you have problems with another, hence the precautionary landing.

Or the other three, in the case of Lockheed's finest. :O

Normally the loss of a hydraulic system will involve some degradation in capability depending on which system is lost, demands are usually spread out so that the loss of a particular system won't result in a complete loss of function. For example, the engine thrust reversers are generally supplied by different hydraulic systems on each side so in the event of a failure one reverser will still be available.

Critical parts, such as the rudder will have dual or even triple sources of supply as they have to work.

The decision to return or divert will be influenced by the destination airport, for example a short runway wouldn't be compatible with reduced stopping performance and not having any maintenance available could result in a prolonged grounding of the aircraft.