One can think of a Möbius strip independently of any particular embedding inside $\mathbb{R}^3$, IE just as a two dimensional manifold. One definition is the set $\mathbb{R} \times [0,1]$ with $(x,0)$ glued to $(-x,1)$ for each $x \in \mathbb{R}$. A bunch of copies of $\mathbb{R}$, one for each element of $[0,1]$, glued to together at the ends with a reversing twist, if you will. Let $M$ be the Möbius strip defined this way. The first nice property of this definition is that every point is the same in the sense that, 'locally' everything looks like $\mathbb{R}^2$.
In the construction above you can use an open interval $(-1,1)$ instead of $\mathbb{R}$. But if you use a closed interval $[-1,1]$ as an alternate definition you don't really have a Möbius strip, but a related object, a 'manifold with boundary'. These aren't quite so nice: nearby every point it may look like $\mathbb{R}^2$, or for boundary points it may look like the closed upper half plane. Manifolds with boundary are not as easy to work with.
The Möbius strip is a good first example of a (non-trivial) vector bundle. Another example is the cylinder, which is what you get from same construction above, but without the twist. While the cylinder is a vector bundle, it is a trivial one: it's essentially just the Cartesian product of a circle with a vector space (the real number line).
For a better info on vector bundles see: http://en.wikipedia.org/wiki/Vector_bundle.