For the maximum matching problem, we can find the fractional matching which I understand involves some sort of weighting for the edges. However, I cannot seem to find an exact and simple explanation of what a fractional matching is. How does it compare to an integral matching?
If this question seems too basic, could I please have a link to somewhere that explains it?
Given a graph $G=(V,E)$, we can represent a matching as a function $f$ from the edges $E$ to $\{0,1\}$ such that for each vertex $v\in V$, we have $\sum_{w\in N(v)} f(v,w) \leq1$, where $N(v)$ is the neighbourhood of $v$, i.e. the set of its adjacent vertices. (We have equality for a perfect matching) In this representation, $f(e)=1$ means the edge $e$ is part of the matching.
A fractional matching can then be represented by a function $f'$ from the edges $E$ to the continuous interval $[0,1]$, with the same constraint, i.e. $\sum_{w\in N(v)} f'(v,w) \leq1$. So, intuitively, each vertex is 'divided' over its incident edges such that it is participating in at most one edge 'in total'.
To add to Discrete lizard's answer, I would recommend you look into mathematical programming and optimization. The matching problem can be modelled as what is called an integer program (in fact the constraints that $\sum_{w\in N(v)} f(v,w) \leq1$ for all $v \in V$ are the constraints that define the matching problem where for each $e \in E$, $f(e)$ is a variable. Furthermore, an integer program demands that the variables be integers. But you can see a natural relaxation of the integer program into a linear program by allowing your variables to take non-integer values. Solutions to this relaxed optimization problem are what we call fractional matchings.
A lot of problems on graphs can be modelled as integer programs, and relaxing them to linear programs is a common technique, so it might be worth looking into.
The formal definitions are very nice, but here's a simplier more intuitive explanation. In a fractional matching, every edge has a number. The sum all all the edge numbers connected to any vertex must be less than 1.
