What is an example of a $n$ dimensional manifold $M$ which is not a lie group or $S^{7}$ but satisfies the following property?:
There is an $n$ dimensional sub vector space $V\subset \chi^{\infty}(M)$ such that every $0 \neq X \in V$ is a nonvanishing vector field on $M$.
So this is a motivation to define an invariant of manifolds as follows: $$ \text{The maximum number $k\leq n$ with a k dimensional subvector space$$ $V\subset\chi^{\infty}(M)$ with the above property} $$
These manifolds, which have a trivial tangent bundle, are called parallelizable. They include all open sets of $\bf R^n$, thus I assume that you had in mind closed (compact without boundary) manifolds.
There are still many closed parallelizable manifolds, the most famous being perhaps all three-dimensional orientable manifolds (Stiefel's theorem, see Milnor-Stasheff, Characteristic classes, Problem 12-B page 148). The simplest example is the product $S^1\times S^2$.
Is this really a vector space?
Suppose it is. Consider a nonzero element $X$ of it. It's negative $-X$ is also there. Perform a tiny perturbation, contained in a small ball: $Y = -X + \varepsilon Z$, $supp(Z) \subseteq B_\delta(p)$. Then the sum $X + Y = -\varepsilon Z$ is not zero and vanishes everywhere outside of the ball!
