In simple intuitive terms, the immersion of a topological object is a reasonably smooth injection or mapping of it into some containing space, while its embedding adds the stiffer condition that the mapping be bijective, i.e. point-to-point with no overlaps.
As a simple example, a topological disc can be embedded in the Euclidean plane but a Moebius strip cannot because it must have a crossing point where the mapping is not bijective. Such an injection of the Moebius strip or, say, a folded disc, is an immersion but not an embedding. The Moebius strip can be embedded in Euclidean space.
Similarly a Klein bottle cannot be embedded in Euclidean 3-space because it must have a crossing line, while the projective plane cannot because it must have a triple-point where three regions of it intersect. Both can be immersed in Euclidean 3-space and both can be embedded in Euclidean 4-space.
Topologically they are, like the disc and Moebius strip, 2D manifolds or surfaces. However where the disc and Moebius strip are manifolds with a boundary, the Klein bottle and projective plane have no boundary. The projective plane in particular is often regarded as a 2D space in its own right. It is only when one tries to embed them locally into some containing space that they become "4D." The only reason we might think of them as 3D is because we tend to think in 3D and they are therefore often first presented to us in textbooks in the form of their 3D immersions.