A rooted device refers to the process where the actual user has access to the system partition and can modify files that belong to the operating system. This is good and bad at the same time because if you don't know what you're doing, the phone might be bricked. Having root access gives you the possibility to remove bloatware (apps installed by carriers like Orange, Vodafone, T-Mobile, etc).
Some phones are equipped with protections against root and rooting your device might and will void your warranty.
You can install any app on the Play Store that comes after you searched for root checker.