While I'm not 100% certain what you are talking about, there are two concepts which you may be referring to:
- Responsive Web Design (RWD)
- Adaptive Web Design (AWD)
RWD is when the page changes on the client side when the window resizes (actually, it can refer to a number of concepts, but the window width is a very common one).
AWD is when the page is served up based on the device detected, typically using the user agent string that gets sent to the server with the web request. It infers a device width and serves up content appropriate for the device.
It sounds as if what you are doing is similar to AWD, though maybe you are redirecting to specialized pages rather than just serving up different content on the primary URL. Don't do that. In fact, I wouldn't recommend AWD at all, as it is inflexible and won't accept new form factors that are introduced in the future (e.g., some new iPad with 10x the resolution of the current one).