Algo is short for Algorithm in the context of Algorithmic trading.
In mathematics and computer science, an algorithm is a set of is an unambiguous specification of how to solve a class of problems. Algorithms can perform calculation, data processing, automated reasoning, and other tasks.
In trading an "Algo" is a set of rules and logic how to automatically perform trading. It is usually programmed into a system that is connected to the markets.
See: Algorithmic trading