Just to clarify, if you have a wi-fi capable device, you will at minimum usually be able to access a store account and download content. What other web-enabled functions you may be able to do will vary from device to device.
I own a Nook Simple Touch Reader (aka Nook, NST, or N2E). Out of the box, you can only access B&N's store to download content via the wi-fi. And yes, you can root it (or so I'm told) to expand those capabilities, but I'm talking out of the box here.
So with the Nook, wi-fi but no web browsing or other internet activities.
If you're shopping for a device, it's best to research its particular capabilities and specs, as they vary.
Hope that helps.