You’ll find plenty of charming inns and hotels on Catalina [1]; most sit in or near Avalon.
You can also camp on Catalina [2], and eco-tourists often prefer to immerse themselves in the natural world of the island by sleeping and eating outdoors.
Links:
[1] http://www.moon.com/destinations/california/los-angeles-and-orange-county/catalina-island
[2] http://www.moon.com/destinations/california/los-angeles-and-orange-county/catalina-island/accommodations/camping