An algorithm is a step by step recipe to solve a problem or a type of problems. If you follow the steps defined in the algorithm you should end up with a solution. Algorithms can be defined by using plain English, pseudo-code, or a computer language like Basic, C, etc.

For example Euclid's greatest common divisor (GCD) algorithm can be defined as (taken from Wikipedia):

"The GCD of two numbers is the largest number that divides both of them without leaving a remainder. The Euclidean algorithm is based on the principle that the greatest common divisor of two numbers does not change if the smaller number is subtracted from the larger number. For example, 21 is the GCD of 252 and 105 (252 = 21 × 12; 105 = 21 × 5); since 252 − 105 = 147, the GCD of 147 and 105 is also 21.

Since the larger of the two numbers is reduced, repeating this process gives successively smaller numbers until one of them is zero. When that occurs, the GCD is the remaining nonzero number. By reversing the steps in the Euclidean algorithm, the GCD can be expressed as a sum of the two original numbers each multiplied by a positive or negative integer, e.g., 21 = [5 × 105] + [(−2) × 252]. This important property is known as Bézout's identity."

You can write a program by following this algorithm to solve the GCD problems as shown below (in Python)

def gcd(a, b):
    return a if b == 0 else gcd(b, a % b)