"Words With Friends" is a Scrabble-like game available for many platforms.
As someone that has played in the past (but quickly gave up after days would pass between turns), I recall you had an option to either sign in with the app itself or use a Facebook login... I believe mostly for contact/friend matching, sending invites, polluting your Wall with game updates, etc.
If you're unaware of your password, perhaps you had it linked to a Facebook account before an app update was applied, logging you out?
There are both free-to-play and paid versions, but the former is supported with ads, as you've noted.