At what level does Skitty evolve?

Skitty does not turn into its second evolution based upon level. It is among a handful of Pokémon that evolve only after being exposed to a certain item or after a particular action is taken.

Skitty can evolve at any level after being exposed to a rare in-game item called a Moon Stone. After having a Moon Stone used upon it, Skitty immediately evolves into the Pokémon with National Pokédex number 301, Delcatty. Delcatty has no further evolutions. Because Skitty's evolution is not based on level, it can be transformed into a Delcatty immediately after it hatches, even if it has zero experience points.