Hello,
I'm afraid your question cannot be fully answered without more details on your phone, its android version,...
On most phones you can capture a snapshot of the current screen . On sony phone you need to hold power and volume down for 1 or 2 seconds.
Now on Lolippop, screen recording (as a movie) has been enabled so if you want to record continuously what is displayed on your screen (provided you are on lollipop) you can install applications like "AZ Screen Recorder" or "Lollipop Screen Recorder".