There are many ways to tell if you have insomnia, but the most common are:
Trouble getting asleep or feeling like you've been lying in bed for hours
Waking up during the middle of the night repetedly and having trouble getting back to sleep
Waking up really early, or sleeping in way late
Feeling drowsy and tired during the day
Feeling a bit sick.
There are other symptons, but they are not as common and you are less likely to have those symptoms. Insomnia can be a short-term nuisence if handled properly, but if not, it can result in long-term problems. So it is best to try and stop the insomina before it happens, or if you already have it, fix it as quickly as possible.
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REM sleep is when you do most of your dreaming. Moreover, lighter REM sleep dreams are those that make sense and can be remembered for a long time. So, one way to tell if you're getting REM sleep is if you're dreaming. You might also choose to have a Polysomnograph, commonly known as a sleep study. A trained technologist can tell if you're getting REM sleep from the readings provided by the various sensors they apply to your skin.
I used to think "I haven´t slept all night" and feel tired all the next day. Now I found WakeApp, an App for the iPhone and it proved me wrong. It monitors my sleep and wakes me up when I´m in a light sleep phase. That way I feel much better and not as sleepy and grumpy in the morning. It shows me how much I really slept - more than I had thought... And the best: it records noises, so I found out, that my fridge is such a noisy thing and it was "keeping me awake at night". So I bought a new one and now I sleep far better.
I can't.
You, on the other hand, can: if you wake up in the morning without outside aid (alarm-clock, your mother yelling at you), then you're getting enough sleep.