As Johnny said, making an offer on an REO property is not different that making an offer on any other property...except that I will add that there are a few more expetations from the bank.
First, they have a lot of properties to sell and they are not interested in messing around. You will most likely have to send proof of funding and proof of earnest money with the offer.
Also, you will have to accept the property AS-IS without warranty. Don't expect that the bank will fix anything or allow you to negotiate upon an inspection. They are not interested in admitting any knowledge of the problems with the property. They will also not allow you to turn on the utilities for further inspection - take it AS-IS or go away.
DO NOT be afraid to make "low-ball" offers on bank owned properties. Many will be reject, but you can only pay what you can pay. That is not to say that you should make it a practice to just make a blind low-ball offer - make sure that you make an offer that is good for your needs.
Banks usually have a policy to not negotiate. You will find that they might counter offer you once on an offer, but they will not usually engage in a long negotiation process. In fact, they will usually just reject an offer outright if it doesn't meet their needs. If your offer does meet their needs, they will get back to you. The best situation is when the agent calls you back and tells you "We are in multiple offers, please present your highest and best offer". What they are saying is that they are interested in getting rid of the property and you are close. Just up your offer by a little and tell them that is the best you can do. They will either accept or reject.
Don't give up on bank properties. The bank has needs and goals and you don't know when those needs and goals change. You can find yourself in the right place at the right time if you just keep offering.