What exactly is Boost?
Boost is a collection of useful and extremely high-quality libraries for C++ that complement the rather small standard library.
What are the most import reasons to use Boost?
Boost offers high-quality tools that are missing from C++. Their use is extremely varied though so whether Boost is for you depends entirely on your needs. But I can safely say that every large enough C++ code base would benefit from using Boost.
Some of the most versatile parts are the shared_ptr (a reference-counting smart pointer that helps prevent memory leaks in pointer-rich code), array which provides a very convenient wrapper around C-style arrays of fixed size and other small odd bits which have been integrated into the next C++ standard.
Is it fully cross-platform?
Almost always yes. This is one of the main qualities of Boost.
Is there any link to a page describing all the modules of Boost in one or two sentences?
There is indeed.