quantum field theory - What does it mean to "wrap" a D-brane around some manifold?

I am getting quite confused with this terminology when I read the papers. Like while constructing the near horizon $AdS_3$ in the $D1-D5$ system one considers $IIB$ on $R^{1,4}\times M^4 \times S^1$ and one "wraps" $N_1$ D1 branes on the $S^1$ and $N_5$ D5-branes on $M^4 \times S^1$. What does it exactly mean?

Coming from reading how D-branes were introduced in Polchinksi's book I would think that in a $9+1$ spacetime Dp branes are some planar streched out stuff with a $p+1$ worldvolume and whose $p$ spatial dimensions are transverse to the $9-p$ spatial dimensions which have been compactified and T-dualized. So are we now saying that its possible that instead of imagining the Dp branes as some set of periodically arranged planes on the T-dual torus we can also think of their spatial world being compactified on some arbitrary p-manifold?

If "wrapping" is really a choice of topology for the p-spatial dimensions of the Dp-brane then what determines this choice? Is this something put in by hand or does this happen naturally?