In old-fashioned string theory (omitting M-theory) one-dim. strings move through a ten-dim. space-time manifold and therefore create a two-dim. string-manifold. The endpoints of open strings are attached to so-called branes which again are sub-manifolds.
Strings may also be described as one dimensional branes, 1-brane; that's a reference to one dimensional space, no time component as Tom describes above...