Do I need to buy an "unlocked" cell phone (from online) for it to work with my service provider T-mobile?

Unlocked means its had the carrier registry number typed into the phone. A phone without this number will not work on any other network other than the one its registered to. Most phones can be unlocked and as long as the carrier its being transferred to is a compatible system for the phone it can be registered on the new network.

Any unclocked phone would just work fine...but if you are looking for a cheap substitute go for a one that is specifically made for T-mobile...as in that case you will be getting it at a subsidise rate... to answer your query an unlocked phone is one which can work anywhere with any service provider where as a locked one is made specific for a service provider. A locked phone comes cheaper as the service provider adds some subsidy for you wont be able to use it with other service provider and hence till time you use the phone you will be their customer.

It does not have to be unlocked if it was originally made for T-Mobile (which most of them are if you searched for T-Mobile phones). The only reason people unlock phones to sell them on the internet is so they charge more them.