startup: a small business that intends to get big. example: a chain of hamburger shops (of course a chain is only one way to "get big").
Note that because they are distinguished by intention, you could have two factually identical hamburger shops, with one being a small business, and the other being a startup run by a deluded nutter, who crazily envisioned a chain of hamburger shops, when every sensible right-thinking person knew that was silly. But I would call it a startup, because of the intention.
pg thinks that it's rare for a startup to succeed (read: "get big") without funding (I'm not so sure, but I can't think of any counter-examples: google, yahoo etc all had funding).
37 signals advocates organic growth, and this works really well for a small business, such as "just a website that sells license plates". 37 signals themselves seem to have grown significantly, but I'm not sure they would be classed a "big company", but seem on the small end of "medium/small business" - therefore they are not (yet) an example of a startup getting big without funding (I'm not sure of their intention - it could be that they don't want this; i.e. that they aren't a "startup" at all). Joel's Fog Creek software is also small at the moment - I don't know whether he intends to become big, and so whether his company is a startup or a small business.