A hashing algorithm is a fast, one-way function that lets you take a piece of data and convert it to something else (called the hash). Thus, you can compare hashes of two strings to make sure that the original strings are the same. A hash collision occurs when two strings have the same hash, and a collision attack is one where given a hash, you try to come up with a string that has the same hash (thereby violating the one way property of the hashing algorithm)
SHA-1 is a hashing algorithm from early 90s that, over the past decade or so, is no longer considered secure against well-funded opponents.
Today, CWI Amsterdam and Google announced a practical collision attack against SHA-1, publishing two dissimilar PDF files which produce the same SHA-1 hash as proof of concept.