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I have encountered the term differential/pushforward many times in the literature, although I cannot seem to understand just what is meant by it. I still cannot seem to understand the definition of the differential of a multivalued multivariable function

$ f : mathbb{R}^n to mathbb{R}^m $

and its generalization to differentiable manifolds. I have seen many definitions of the differential, particularly those for tangent vectors on manifolds and the definition with derivations of functions and one involving the Jacobian matrix, but I cannot understand this or any of them, thus I also cannot understand just what is meant by tangent space on manifolds. What does the differential "really mean" and how to use it? In particular, how is the differential/pushforward related to derivations of functions? Could someone please explain the differential to me and possibly using it to define tangent spaces on manifolds. I am frustrated as I have never understood the differential and lack the proper understanding in tangent spaces. All help is appreciated.

Let $fcolonmathbb{R}tomathbb{R}$ and $xinmathbb{R}$, then $f$ is differentiable at $x$ if and only if there exists $alphainmathbb{R}$ such that:
$$f(x+h)=f(x)+alpha h+o(h),$$
when $alpha$ exists, it is unique, denoted by $f'(x)$ and called the derivative of $f$ at $x$.

Proof. Assume that there exists $betainmathbb{R}$ such that $f(x+h)=f(x)+beta h+o(y-x)$, then:
$$(alpha-beta)(y-x)=o(y-x),$$
whence $alpha=beta$. $Box$

Remark. Notice that $hmapstoalpha h$ is a linear map.

Geometrically, $y=f(x)+f'(x)(y-x)$ is the best line approximation of the graph of $f$ around $x$.

Now, let $fcolonmathbb{R}^mtomathbb{R}^n$ and $xinmathbb{R}^m$, generalizing the above definition, $f$ is differentiable at $x$ if and only if there exists a linear map $ellcolonmathbb{R}^mtomathbb{R}^n$ such that:
$$f(x+h)=f(x)+ell(h)+o(h),$$
when $ell$ exists, it is unique, denoted by $T_xf$ and called the differential of $f$ at $x$.

Proof. Assume that there exists a linear map $ell'colonmathbb{R}^mtomathbb{R}^n$ such that $f(x+h)=f(x)+ell'(h)+o(h)$, then:
$$(ell-ell')(h)=o(h).$$
Let $hinmathbb{R}^nsetminus{0}$ and $tinmathbb{R}^*$, then $displaystylefrac{(ell-ell')(th)}{|th|}=frac{(ell-ell')(h)}{|h|}$ converges toward $0$ when $t$ goes to $0$, therefore:
$$ell(h)=ell'(h),$$
and this also holds for $h=0$. $Box$

Remark. This definition can be extended to maps defined only on an open set of $mathbb{R}^m$, as $x+h$ would fall in this open set for $h$ being sufficiently small.

I won't explain how to define the tangent space of a manifold and the tangent map of a differentiable function as it is tedious and quite long. However, the main idea is that a manifold is locally an open set of some $mathbb{R}^m$.

I recommend you have a look at An introduction to Differential Manifolds by J. Lafontaine.