The first thing that's likely to happen after you put in an offer on a house is that the seller will counteroffer, perhaps asking for a higher price, or requesting other changes to the terms of your offer (maybe that you drop your contingency to sell your own house first, or that you add a contingency allowing the seller to close on another house before closing on and moving out of yours). You might counter the counteroffer, and so forth, and this may go on for a number of days.
After this part of the process is wrapped up and you've entered into a purchase contract, how long things will take depends on your contract, which should specify a closing date or number of days before closing. Within four to ten weeks is typical, although it varies from state to state.
You, the seller, the escrow agent, and pretty much everyone involved in the transaction will probably be working their hardest to make the closing happen on time. If it gets delayed, however, it's usually not the end of the deal -- you and the seller can simply agree to put it off by a few days. Unfortunately, extra demands by buyers' lenders are currently a common source of closing delays.