If you have a REM cycle, which you should have at least one every night, you are definitely dreaming.

REM stands for Rapid Eye Movement, and it's because you're looking around at things within your dreams. Unlike the rest of the body, the eyes are not typically affected by the natural process of sleep paralysis (the system your body uses to stay still so you're not constantly acting out your dreams in bed.)

Fascinatingly, the brain/sub-conscious naturally purges dream memories as soon as it deems them 'not-reality'. You can train your brain to rememember your dreams more if you write them down as soon as you wake up, this tells your sub-concious that those memories are actually worth remembering.