When people use the term service economy, they are refering to one or both of two recent economic developments. One is the increased importance of the service sector in industrialized economies. Services now account for a higher percentage of GDP than just 20 years ago. If you look at the current list of Fortune 500 companies, more of them are service companies and fewer are manufacturers than in previous decades.
The term is also used to refer to the relative importance of service in a product offering. That is, products today have a higher service compenent than in previous decades.