The BibTex format is, simply, a format for cataloging papers/other documents. If you think about it in terms of a database, it just provides a standard way of laying out the database fields.
Using BibTex means that lots of different citation management programs can read your bibliography.
A BibTex (.bib) file is simply a list of BibTex references one after the other (if you open up a .bib file in a text editor you’ll just see a big list of papers in a standard format).
There are a number of bits of software that will read, manipulate and generally allow you to do nice things to your BibTex files. I think JabRef is great, but we could start a full list of bibliographic software here.
You can also simply paste BibTex references into pages on this site... You can read how to do this.
More information is avaliable on this (and lots of other) sites.