I will avoid calculus mumbo-jumbo as much as I can.

In simple terms, the gamma function is an extension of the factorial so that it can take non integer inputs. This would include complex numbers, something that the factorial can't do. The gamma function can provide a value for any constant such as, pi .9, 2.5, e (Euler's number), ect. An advantage as the factorial by definition can only take positive integers (1, 2, 3, 4...). Also the gamma function, much like the sine and cosine, tends to be found in places that do not seem related at first, this is largely due to it's highly general nature.

The function itself often stated as a integral (if you do not know what that is reading that node would be a good place to start), to compute this you will need the ability solve an improper integral.

See factorial for more details on the general idea of this function.

If you are interested in something very complex see complex number.

get it complex it's a math joke.