An algorithm is a set of instructions for solving a problem or completing a process, often used in math. The steps in an algorithm are very precise and well-defined. If your problem is a headache, your algorithm might look like this:
1) Have you been hit on the head? If yes, seek medical attention; if no, go to the next step.
2) Have you taken a pain reliever? If no, take one now; if yes go to the next step.
3) Have you eaten today?
… and so on until it would end with either a solution or advice to seek medical attention.
Algorithms often take the form of a graph with a square for each step and arrows pointing to the possible directions from each step.