In what one might call naive calculus, for each coordinate $x_i$, the differential $dx_i$ denotes a small (infinitesimal, even) change in $x_i$,
so a covector $\sum_i a_i dx_i$ is an infinitesimal change.
On a manifold, coordinates are only local, not global, so we should also imagine
that this covector sits at a particular point of $M$. If we want to have a covector varying smoothly at every point, this is a differential one-form.
If $f$ is a function, then the total differential of $f$ is the quantity
$$df = \sum_i \dfrac{\partial f}{\partial x_i} dx_i, $$
which records how $f$ is changing, at each point.
A tangent vector at a point is a quantity $v = \sum_i a_i \dfrac{\partial}{\partial x_i}$; you should think of this as a vector pointing infinitesimally, based at whatever point we have in mind. You can measure the change of $f$
in the direction $v$ by pairing $df$ with $v$.
Summary: tangent vectors are infinitesimal directions based at a point, while covectors are measures of infinitesimal change. You can see how much of the change is occurring in a particular direction by pairing the covector with the vector.
Now higher degree differential forms are wedge products of $1$-forms. You can think of these as measuring infinitesimal pieces of oriented $p$-dimensional volumes. (Think about how the (oriented) volume of an oriented $p$-dimensional parallelapiped spanned by vectors $v_1,\ldots,v_p$ depends only on $v_1\wedge\cdots\wedge v_p$.)
Best Answer
Non vanishing (at, say, $p$) means that there is a vector $v$ in $T_pM$ such that $\alpha_p(v)\neq 0$. Similarly for the $k$-form, it means that there is a set of $k$ vectors such the form is nonzero if evaluated on these vectors.