Rickair7777 pretty much has the right of it. Each company tends to do it a bit differently. When I received an invitation to interview at one regional, they emailed a bunch of documents that I had to print out, sign, and send back prior to the interview. Once I was offered a position, they sent additional forms they wanted signed and returned before my class date. I also had to have a drug test conducted between the interview and class date. When class began, they brought in another round of forms, but at this point they were largely related to health insurance, tax witholding, etc. Most of the background check-related forms they'd had from before the interview, but I doubt they did much with them prior to selecting candidates.
One thing to bear in mind is that just when they conduct the background check isn't nearly as important as what it turns up. If you read the fine print on some of the disclosure forms you sign, they say that "an offer of employment is contingent on the results of the background check" or something along those lines. This means that even after you've been offered and accepted a position, any surprises could mean you're back out on the street. It doesn't matter if it's prior to the interview or after you've been on the line for a month. The company still has the right to fire you.
Sign the forms whenever each particular company gives them to you, but be sure you've been forthcoming about anything they could possibly turn up.
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