One of the best things you can do is look in your local paper or online to find out how much homes in the area you're looking in have sold for recently, and if possible, find out what the original listing price was. This should give you an idea of the kind of demand you're dealing with. Ultimately, demand in the area is going to determine whether or not your offer will be accepted (within reason).
For instance, if a seller lists a house for $400,000 and then gets 5 offers, they are in a really good position. Even if you offer the full list amount, someone bidding against you who is familiar with the current market climate might have bid higher than the listing amount. Of course, it there is little demand, you don't want to overbid and be the only offer, because at that point you're just throwing money away.
It gets more complicated when you find a house that is "the one." A lot of times home buyers are so attached to a house that they are willing to pay more (sometimes far more) than the home is worth just to make sure that no one else gets it. It's your money, and you're the one who is going to have to own the home, so we can't really advise you further in this case. But just keep in mind, if you were to bid let's say $390,000 on that same house...that's $10,000 that you can spend on cute furniture and stuff to decorate your new home!
If you have a real estate agent, he or she can probably give you additional information about the home and the current market to help you determine what sort of offer you should make. But keep in mind that how much you pay for the house directly impacts your agent's commission, so it's really in their best interest to get you bidding high.
Check out the following video for more great information about home bidding!