The best Facebook app in my opinion is Friendly, and I bought the paid version, which is worth it in my opinion. You can chat through friendly, and you can also get another app called Chat for Facebook with Push. As long as you have the app running in the background, you will always be connected and online to Facebook, and if someone try's to chat with you, then a message will pop up on the screen. I forgot how much it cost. $1 or $2 I think. Both apps are very good.
Also, Friendly does NOT push chat notifications like the Chat for Facebook app.
And also, there is an app called Flickpad Pro. It basically updates you on any new pictures uploaded from friends on Facebook and flickr (if you use flickr). I think it was $2.99, very much worth the money.