An Advance Reading Copy (ARC) is a small print run of a book that is distributed to a limited group of people before the official print run. This is done for a number of reasons, including to build up some buzz, give reviewers time to get their reviews ready, encourage librarians and booksellers to preorder, and to give one last chance for people to find and point out mistakes.
ARCs tend to be one step away from their final form, and may have a different binding or formatting from the final product. It is common for them to be trade paperbacks, and the cover may look somewhat unfinished, as space is often left for jacket blurbs that will come from the ARC's reviews. ARCs are usually labeled 'uncorrected proof' on the cover, being the next step after the galley proofs used for editing.
ARCs are free, but it's not easy to get ahold of one. As you might expect, you are not supposed to sell the ARC, so some bookstores will give them out as prizes once they are done with them. Some websites, such as NetGalley and Goodreads allow you to request advanced copies; NetGalley with the expectation that you will review the book, and Goodreads in the hope that random giveaways will create buzz.
In some cases ARCs may be valuable to collectors, and in some cases ARCs are even signed by the author. However, first editions are more popular among collectors. An ARC print run is usually about 5,000, which is both very limited but not so limited as to saturate the market for collectors.
A note on terminology: there are a lot of terms for this. I chose Advance Reading Copy because it is both common and because the abbreviation ARC is commonly used. However, the term Advance copy is perhaps more generally applicable, and covers all variations, e.g., advance reading copy, advance review copy, advance reader's edition, and reader's edition.