If you'll forgive a non-American for answering:
The term 'biro' comes from the name László Bíró, the Hungarian inventor. It is an everyday, informal term for a ballpoint pen, used exclusively in Britain. It is almost unknown outside the British Isles , so you are unlikely to find many Americans who are familiar with the term. It's also used by Australians (who share a lot of everyday language with the UK) and to a limited extent in Italy.
Note that we wouldn't use it in formal situations, even in the UK. An official form would be printed with the words 'Please complete in ballpoint pen' - you'd never see 'biro' in such a context.
If I were you, I'd keep this word in your passive knowledge. You need to understand it if you happen to work in an office with British colleagues, but otherwise, it's not much use. Just stick with the universally understood term 'ballpoint pen', and you can't go wrong.