Actually not much like the Shuttle at all
The Shuttle was a fairly conventional body-and-wings design like a normal aircraft.
People go with lifting bodies because they want to minimize the wing are, and hence the area you need to cover with a heat shield.
Otherwise LB's tend to have much worse handling characteristics than body-and-wings (which is an impressive feat, given the Shuttle handled like a brick).
DC is actually the most cutting edge tech seen for ISS resupply. It's a crew rated (given it has to berth to the Station) all composite (carbon fibre) human sized lifting body. Nothing has all those things together in one package.