An equilibrium solution is a solution to a d.e. whose derivative is zero everywhere. On a graph an equilibrium solution looks like a horizontal line.
Given a slope field, we can find equilibrium solutions by finding everywhere a horizontal line fits into the slope field.
Equilibrium solutions come in two flavors: stable and unstable. These terms are easiest to understand by looking at slope fields.
A stable equilibrium solution is one that other solutions are trying to get to. If we pick a point a little bit off the equilibrium in either direction, the solution that goes through that point tries to snuggle up to the equilibrium solution.
An unstable equilibrium solution is one that the other solutions are trying to get away from. If we pick a point a little bit off the equilibrium, the solution that goes through that point is trying to run away from the equilibrium solution.
If the solutions are trying to get away on one side and snuggle up on the other side, the equilibrium is still unstable.
If we're given a differential equation instead of a slope field, we can determine whether each equilibrium solution is stable or unstable by using the differential equation to sketch a very rough slope field.