Ayleid Ruins are big, white Roman-like buildings where an the ancient civilization lived 3000 years ago, before Cyrodiil was taken over by men with Saint Alessia. There are a wide variety of things inside, like good loot, and the enemies vary greatly. Watch out for some of the traps.
The enemies you find depend on the ruin. In Lipsand Tarn, for example, you'll find Vampires. In the ruins of Veyond, you'll find Trolls. In Ceyatatar, you'll find Conjurers. But mostly, you'll find the undead. Ghosts, Skeletons, Liches, the works.
The first one you are likely to find is Vilverin, a Bandit Lair across the lake from the exit of the Imperial Prison Sewers at the start of the game. You can't miss it.
In Ayleid Ruins, you may also come across strange stones. The blue, aqua-marine colored ones are called Welkyend Stones. These are the most common, and they also restore all your Magicka. Very useful for those mages, or people with the Atronach Birthsign. The white ones are called Varla Stones, and they restore all charge to your magic items.