As Scott said, logos can be used at any size from 50px to 50ft. However, in my experience, most people tend to design a logo with the printed size of 2"-6" in mind (in either width or height, generally centered on an 8.5x11" paper). Some people are very gifted at designing for extremely large and small scales, but for the rest of us, here is a rule of thumb that I tend to use:
- Reproduce the logo at 1/2"-1". In most cases, this will be the smallest size used for stationery, at the corner of a website, etc. (although I have seen smaller logos like 1/4" used on printed materials before). At this size, you will be able to see if any aspects of the logo are lost at smaller scales.
- Reproduce the logo at 2'-3'. When viewed up close, this will give you a good idea of the logo's legibility at a large size, or if any parts of the design overwhelm or are difficult to discern at a larger scale. Viewed from a distance can actually reproduce how a billboard would look when viewed from the ground (this trick can be used to judge the legibility of any billboard designs).
However, for any logo you design, you should have a clear idea of its usage - vehicles, billboards, signs, stationery, digital reproductions, etc. - but these two tests will allow you to check the logo's versatility, especially at very early steps in the design process.
Whether you would need a single or multiple logos really depends on your design. It is possible to have a single logo that is easily legible at all sizes (think the FedEx logo), but some more detailed logos may make use of a single icon or distinctive letter from their full logo as a small-size logo.