There are a ton of options out there, from CoreData to SQLite to Yap, but it's hard to see what's commonly used these days for good iPhone apps. Bonus if the database supports syncing to a server we control (not iCloud) and has good bindings in other languages (like JS or Android).

We use (and pay for) Parse at the moment but it's really slow for us despite the very little amount of data we send to them and caching locally is a big pain.