Take a simple differential equation as an example, like dy/dx = x. To solve this we need to find a function y(x) whose derivative is equal to x. This particular differential equation is well-behaved, and the solution is simply 0.5*x^2, which you can verify using basic calculus.

Not all differential equations are nicely behaved. Many do not have analytic solutions at all, meaning that we can't even write down a function that satisfies the equation. However, such equations can still be solved approximately using computational techniques. This can be incredibly difficult and computationally expensive (note the huge resources required to achieve this breakthrough) but can produce very useful results.