Serena has accurately described the short sale process, and that's likely the route you would/should take. You make an offer and provide some very specific documentation; the offer is sent to the seller's lender (again, with some very specific information--hardship letter, copies of past tax returns, W-2s, etc.), and then (with some negotiating) it's up to the lender to decide whether to accept, reject, or counter your proposed purchase price.
Again, a short sale is the "traditional" way of buying a property not yet in foreclosure. However, just for fun, here's another: Buy it "subject to" the existing financing. Here's what you do and here's what happens. The owner of the property transfers ownership of the property to you; you receive the deed. You, in turn, make up any current deficiency (any late mortgage payments, penalties, interest) and you promise the seller to make the future mortgage payments. At some point, you will get new financing on the property in your own name.
Advantage to you: You own the property right away. And you don't need to get new financing on the property. Disadvantage to you: You very likely will be violating the lender's "due on sale" clause. Worse case scenario: The lender could require you to refinance the property immediately. That doesn't happen often; usually the lender doesn't know or doesn't care. All the lender cares about is that any deficiencies are paid and that future payments are coming in on time. Also, you have to come up with enough cash to take care of back payments.
Advantage to the home owner: They quickly have a resolution to their problems. Disadvantage: A major one: They've transferred the ownership of the property to you, but they're still legally on the hook for the mortgage. So they want to make sure that you'll keep paying the mortgage until you refinance the property in your own name.
There's also a technique using a land trust that accomplishes something similar. Ownership of the property is transferred to the land trust, with the seller and you both as beneficiaries to the trust. The trust documents specify the details of the transaction. A land trust actually protects the buyer and seller more than a "subject to," and eliminates the "due on sale" problem. However, don't try doing a land trust yourself; properly constructed, the documentation can run 60-80 pages or more.
Hope that helps.