Mumford studied the quasi-projectivity of moduli space of vector bundle. But such big family is unbounded. So he needed to consider moduli space of vector bundles with some sort of GIT. He arrived to the notion of slope stability. One advantage to restricting to semistable bundles of fixed rank and degree is that
the moduli problem is bounded. In fact Mumford extended Giesecker-stability notion. See [section 2.5][1] 


We have nice interpretation between the notion of semistability for vector bundles and the notion of
semistability coming from an associated GIT problem 







We can interpret the slope of vector bundle over a curve of genus $g$ using Riemann-Roch formula. Let me explain it


From of differential geometric point of view the degree of a holomorphic vector bundle can be computed by Chern–Weil formula in terms of curvature, and the fact that curvature decreases in sub-bundles. We explain Chern-Weil formula which gives an effective way for degree of holomorphic vector bundle.

A reflexive sheaf (i.e its double dual is equal itself) is locally free (i.e., a holomorphic vector bundle) outside a subvariety of codimension greater than or equal to two. Let $\mathcal F$ be a coherent subsheaf of holomorphic vector bundle $E$, then there is an analytic subset $S \subset M$ of codimension bigger than two and a holomorphic vector bundle $F$ on $X \setminus S$ such that
$$\mathcal F|_{X\setminus S}=\mathcal O(F)$$
and $F$ is a sub-bundle of $E|_{X\setminus S}$ and there is an orthogonal projection $\pi:E|_{X\setminus S}\to F$
which $\pi\in L_1^2(End(E))$ lying in the Sobolev space of $L^2$ sections of $End(E)$ with $L^2$ first-order weak derivatives and satisfying $\pi=\pi^*=\pi^2$ where $\pi^*$ denotes the adjoint of $\pi$. The Chern-Weil formula is $$deg_\omega \mathcal F=\frac{\sqrt[]{-1}}{2\pi n}\int_X tr(\pi\Lambda_\omega F_h)\omega^n-\frac{1}{2\pi n}\int_X|\nabla''\pi|^2\omega^n$$
where $\nabla''\pi$ is computed in the sense of currents using the $(0,1)$ part of the Chern connection of $E$.



We define the slope of $\mathcal E$, to be $$\mu(\mathcal E)=\frac{deg \mathcal E}{rk \mathcal E}$$


For any non-trivial vector bundle $E$ on curve $X$ by using the Riemann-Roch formula we can compute the slope of a vector bundle over a curve as follows,

$$\mu(E)=\frac{dim H^0(X,E)-dim H^1(X,E)}{rank E}+g_X-1$$
where $g_X$ is the genus of curve $X$.

Another interpretation about semi-stability via slope 

It is known that a holomorphic
line bundles on a compact connected Riemann surface $\Sigma_g$ 
do not admit non-zero global holomorphic sections if their degree is negative.

A non-zero homomorphism from line bundles $L_1$ and $L_2$(which is a section of the line bundle $L^∗_1 ⊗ L_2$)  exist if $deg (L^∗_1 ⊗ L_2) ≥ 0$, which is equivalent to $deg L_1 ≤ deg L_2$


Note that, for higher rank vector bundles, the degree of $E^∗_1 ⊗ E_2$ is
$$deg (E^∗_1 ⊗ E_2) = rk (E_1)deg (E_2) − deg (E_1)rk E_2$$

so the semi-positivity  condition is equivalent to
$$\frac{deg E_1}{rk E_1}
≤
\frac{deg E_2}{rk E_2}
$$



  [1]: https://ncatlab.org/nlab/files/SaizStableBundles.pdf