There's plenty of sites which will let you host images for free. One way would be to look at the addresses of other peoples' sigs to see what they use for hosting.
BBCode is basically a forum-safe version of HTML, it uses [ and ] instead of < and > for the tags, and has less tags so people can't use the malicious ones like EMBED or IFRAME etc. (unless HTML is also enabled). I'd expect most boards to at least support [b], [i] and [u], though some may support more as well, like this place does - you'll have to read FAQs etc. to see what's available at different places.