An algorithm is simply a set of instructions or rules that describe how to perform a task.  Whether you know it or not, you already use algorithms in your daily decision making.

In computer science, these instructions are based on logical rules, which makes them interpretable by a computer.

In machine learning, these instructions are specifically designed to efficiently find patterns in data.  They are special in that their performance improves over time without any additional explicit programming.

Consequently, these algorithms are quite flexible, and therefore can be applied to unique problems on unique datasets with only slight modifications.

The final result of applying an algorithm on training data is a “model”.  The trained model can be used to predict the outcomes of unseen examples.

“Under the hood” this is accomplished through a combination of basic mathematics, statistics, probability, calculus, and linear algebra.

This math allows the algorithms to find an optimal solution in the least amount of time possible.  Without this, most modern machine learning would be impossible, even on the most powerful supercomputers.