Boost is a collection of C++-libraries. The bar to be accepted into Boost is very high, so most libraries in Boost have a very high quality, both by design and the actual implementation.
Many libraries from Boost have made it into C++ standards. Some examples include smart pointers (unique_ptr, shared_ptr, weak_ptr), thread and chrono. In the future ASIO will probably be a part of a standard too.
The number of libraries keeps increasing. At the time of writing, there are about 145 libraries available. Not all are used by everyone, but some of those are useful. Other libraries are for C++ professionals and some are for experts in other domains.
The advantages with Boost are tremendous, but there are a couple of backsides to the medal. Each release is heavy, and it contains easily about 55000 files, and takes about 500 megabytes of space. If you use a library which have to be built, the disk cost can increase with 5-10GB.