What is the bandwith limit?

There are two different limits:

1. The limit of raw data that can be downloaded from the API. This API is meant for edits and thus limits the amount of data that you can download to a amount that you wont normally reach while mapping. If you need more data have a look at planet dumps (also available as regional extracts) or the XAPI. If you exceed that amount you will be automatically blocked with the message "bandwidth exceeded".
2. The limit of renderd tiles that you can download. What is and isn't fair use is laid out in the tile usage policy. This is counted per source connection. If one user (or one application) download so many tiles that it is noticable by a server admin it will be served the yellow-black tile shown on the wiki page. If 100 of your users view your website with openlayers and openstreetmap tiles embedded, everything should be fine. If you have 100000 users viewing your website and requesting tiles we will probably notice the referrer, contact you, tell you to stop it and if you don't, we'll block you manually.