A damp patch on your shirt isn't anything to worry about. It could be an external source of moisture (like spilling something on yourself), or the moisture could be coming from within, like sweat. You can wait for it to dry, or you can change the shirt, but it's not exactly a big deal. Similarly, a damp patch on your plaster walls could have an internal or external source. Some accidentally-spilt water on a plaster wall isn't going to require any attention and will dry of its own accord (dabbing it with a paper towel is optional). It could also be a habitual problem, like regularly hanging laundry to dry inside on clothes rack which continually brushes against the wall. Change this habit, and the problem will go away. It's when a damp patch on your plaster wall has an internal source that you need to be concerned.
Areas of Discolouration
Damp patches on a plaster wall initially present themselves as small areas of discolouration. These areas can grow darker the longer they're left unrepaired, and in cases where the area has become sodden, the plaster can even bubble. Where is that moisture coming from?
There's no best-case scenario with damp patches on a plaster wall, but there are better-case scenarios. Hopefully, the issue is localised, such as a minor leak in a pipe directly behind the damp patch. A plumber can quickly sort this out for you. A more problematic explanation is a leak in your roof, allowing water to make its way downwards. Neither of these issues will go away of their own accord, so you will need professional repairs to cut off the source of the moisture accumulation.
Reversing the Damage
Small leaks where the damp patches are barely noticeable can largely dry out by themselves (once the source of the leak has been identified and patched), but you could still be left with areas of discolouration. With larger leaks, this is all but a certainty. The plaster is damaged and will need to be replaced. Not only does the section look unsightly, but it's now weaker, and can be more easily prone to damage through usual wear and tear. A professional plastering team can generally perform the replacement work in a sectional manner, meaning the damaged part is cut out and replaced without needing to replaster the entire wall, although it will usually be necessary to give the wall a fresh coat of paint.
These damp patches on plaster walls need to be addressed as soon as they're detected. Water erosion formed the Grand Canyon, and you don't want an accumulation of moisture to erode your home.