The generalisation of calculus from vector spaces to manifolds relies on a crucial observation, namely that elements of vector spaces have two properties at once: they are both a "vector" and a "point". So what is meant by these words? Loosely speaking, a vector stands for a direction whereas a point indicates a location.
You can see this in the following example. Consider a (smooth) function between (finite-dimensional) vector spaces $f:V\rightarrow W$. Denote its derivative by $Df$. Then the directional derivative of $f$ in direction $v\in V$ is defined as
$$
D_vf: V\rightarrow W, \quad p\mapsto D_vf(p):=Df_p(v)\equiv Df(p)(v)
$$
So $v\in V$ denotes the direction in which we want to differentiate, whereas $p\in V$ is the point where we want to evaluate the derivative.
A first abstraction is done by generalising this concept to affine spaces. An affine space is a tuple $(A,\vec A)$ consisting of a set $A$ (the "points"), a vector space $\vec A$ (obviously the "vectors") and a mapping $+: A\times \vec A \rightarrow A, (p,v)\mapsto p + v$ ("the translations"), which works exactly as in vector spaces (for details see Wikipedia).
Then the derivative of a function $f$ between affine spaces $A$ and $B$ is defined as in vector spaces (we just have to replace addition with translation). It now represents a map
$$
Df: A\rightarrow \operatorname{Hom}(\vec A, \vec B), \quad p \mapsto Df_p,
$$
i.e. $Df_p$ is a linear map between vectors in $\vec A$ and vectors in $\vec B$.
Especially, if $B=\mathbb R$, then at each point $Df=df$ is a linear functional on $\vec A$ which makes $df$ a differential 1-form on $A$. In differential geometry, differentials are thus seen (equivalently!) as mappings from vectors to functions
$$
df: \vec A \rightarrow C^\infty(A), \quad v \mapsto df(v).
$$
At this point you probably already understood that there is a difference between points and vectors on the level of affine/vector spaces. I will now say some words on how these concepts are carried over to manifolds by differential geometry. However, I will only present some ideas. If you are interested, you should attend either a lecture or have a look in a book (e.g. J. M. Lee, "Introduction to Smooth Manifolds").
Another point of view is to think of the vectors $v\in\vec A$ as to be attached to points $p \in A$, formally we think of the product $A \times \vec A$. Then a pair $(p,v)$ can act on smooth functions by the directional derivative:
$$
(p,v): C^\infty(A) \rightarrow \mathbb R, \quad f \mapsto df_p(v)
$$
The idea of differential geometry is to take this point of view and make this concept local. I.e. allow the directional space $\vec A$ to vary from point to point. More precisely, the above concept is even taken as the definition: The directional space at $p$, the tangent space, is defined to consist of all such functionals $(p,v)$.
Note that the formalisation of this idea is a lot more work since you first have to define manifolds, declare a smooth structure (to know what a smooth function is) and then construct the tangent spaces based on the idea above.
