It really depends on what exactly you plan on using the laptop for. For instance, if you just want to check your e-mail just about any laptop will do, but if you want to download large files, edit movies, play games, the requirements are much more demanding. In short, make a list of what you expect out of the laptop, as well as the price range (they can cost a few hundred to a few thousand dollars) and we'll make some suggestions.
In regards to connecting to the internet, that depends on how you wish to connect:
1.) Dial-up modem, which is standard even on older models.
2.) Broadband (DSL/cable)...requires an Ethernet or USB port on the laptop.
3.) Wirelessly...basically requires the laptop to support 802.11b/g.

Of course you can usually purchase a card for the laptop to add the ability if it doesn't have it already. (Note: Just remember that you must pay for monthly internet access.)
Personally, I'd be wary of anything on Ebay, particularly refurbished units, because there are so many things that can be or could go wrong with it...not everyone is as honest and trustworthy as you'd like them to be. However, if you're looking for a cheap laptop then it may be the way to go.

Fill in the details of use and cost and we'll help point you in the right direction. Also, feel free to post a couple links and ask for feedback if the computer's a good deal or not.