RegionalExpress (short: RE) is Deutsche Bahn's designation for state-ordered, midrange commuter trains that will not stop at every station of a line. (It is possible that other German railway companies are running trains called RegionalExpress, too; I haven't checked.)
Due to Germany's strange dualism between "long-range" and "short-range" rail traffic, where long-range trains are run on a free-enterprise basis and short-range rail traffic is ordered by local linked transportation systems, which are constituted by Länder and communities, DB Regio's RegionalExpresses may compete with DB Reise&Touristik's InterCity, EuroCity or even InterCity Express trains on the same lines.
The rolling stock constituting RegionalExpress trains varies. On electrified main lines, it ideally means some medium electric locomotive as a 111 or 146 plus several double-decker cars, but REs consisting of old, more or less renovated 'Silberling' cars or PUMA/MODUS cars also happen.
On non-electrified lines, high-turnover connections are run with a 218 diesel locomotive plus double-decker cars (those with windows you can open, as the new air-conditioned double deckers would suck up the diesel fumes). On other connections, DMUs such as the 612 are usually employed.
The latest development in the RegionalExpress' history was the introduction of RE160 relations, where regional trains go as fast as the conventional German signalisation system allows.