Given some fixed coefficients $a,b,c,d,e,f$ the fifth degree equation refers to
the polynomial of degree $5$,
$$
f(x) = a x^5 + b x^4 + c x^3 + d x^2 + e x + f
$$

The problem now is to find the exact solutions of the equation

$$
f(x) = 0
$$

in terms of $a,b,c,d,e,f$. A classical question in mathematics asks if it is possible
to express these solutions by "radicals", that is, using only the operation of addition, subtraction, multiplicative, division, and taking radicals (i.e square-roots, cube roots, and so on) applied to $a,b,c,d,e,f$.

The answer to this question is known. It is not possible to solve the equation $f(x) = 0$ by radicals when $f$ is a degree 5, or higher, polynomial (as in this case), but possible for polynomials of degree four. This solution comes from a field of mathematics known as Galois theory (with previous work by Abel).

Now enter the elliptic function. The elliptic functions are a class of function with very rich properties and that have been absolutely central in the mathematics of the 19-th century. They still hold an important position to this day.

The relation between the equation $f(x) = 0$ and the elliptic function is, I believe, that while it is impossible to solve the equation $f(x) = 0$ by radicals (i.e in terms of these 5 or 6 operations applied to $a,b,c,d,e,f$), it is possible to express the solutions to the equation $f(x) = 0$ in terms of these 5 or 6 operations and elliptic functions.

I don't know much more beyond this point. If I did say something wrong feel free to correct me.