Andriod is an operating system that "runs" the phone. In the same way that iOS is the operating system that runs the iPhone (or the same way Windows runs your PC).
As far as I know any phone that runs Andrioid is a smartphone.
If you want to use the internet you need a smartphone and an Andrioid phone will do it nicely (and cheaper) than an iPhone.
There are many Andriod phones nowadays, from many companies, and the Samsung Galaxy S2 is an excellent phone and does very well in tests in magazines.
The Samsung Galaxy Nexus is the latest Andrioid smartphone, running the latest version of the Andriod operating system, though as it is very new it is unlikely to be cheap.