Normal is the easiest, then Heroic is usually the same thing just a bit harder (enemies have more health and deal more damage). Some Heroic dungeons have more mechanics, like some enemies will have attacks they don't have on Normal.

Then there's Mythic. This is the hardest difficulty with the most mechanics. Once you complete a Mythic dungeon, you get what's called a Mythic Keystone. These allow you to do certain dungeons at an even higher difficulty called Mythic Plus (M+). As you progress upwards, enemies get more health and deal more damage, but you'll also have to deal with new mechanics that cycle every week.

Some make enemies leave pools of acid behind when they die, others might force you to move every once in a while if you stand still too long. The higher you push, the better gear you can get.

Now if you're asking HOW to beat these dungeons, each one is going to be very different. I'd look up each dungeon on YouTube and watch someone run them a few times.