Escrow is a broad topic but the basics are that a buyer and seller agree upon a trusted third party to handle the funds and processes of the transaction. The buyer will typically put an earnest deposit into an escrow account. An escrow/title agent will walk the buyer and seller through the process. To "open escrow" is to get a property under contract and to start walking towards a transfer. Upon acceptance of an offer generally the buyer's agent will send the contract and related documents and earnest deposit to a title/escrow officer of the buyer's choosing.
Generally on residential properties the buyer has 10 days to do due diligence (inspections and the like) to make sure they discover everything they want to uncover. If the buyer finds something not to their satisfaction, they can elect to cancel escrow (during the 10 days for just about any reason) and get their earnest deposit back. I had this happen just a few weeks back. Didn't like the information I uncovered and canceled escrow.
The escrow/title agent works with both buyer and seller as well as the lender, HOA, and any other parties involved in the transaction. They will prepare the settlement statements as well showing who is paying/receiving what monies. They will also sell title insurance and run a title check to ensure the title is clear. Generally people use a title/escrow service for the insurance and safety factor but it's not always the case.
Title/escrow officers receive a fee for this service.
Once the funds come in from the lender, the escrow agent will record the deed with the county and disburse funds to the seller.
There are nuances and things that can come up but in general that's how it works.
"Escrow" has more than once meaning, it depends on what you are talking about.
Either it is the closing agent, aka Escrow Officer, as Justin discussed or it is the account held by a lender for the impound account. All too often, people think I am holding their impound funds, which we do not and that is like comparing apples and oranges. :)
Escrow is not the word used for the settlement agent in every state, Cali is the only Escrow-Driven state in the country, so it is not likely you will hear "Escrow" as referring to the closing agent in too many states. We are often called a closing agent or settlement agent.