Simply put, Cross means solving the cross on any side of the cube with the edge pieces matching the color of the centers on all 4 other sides. F2L stands for "First two layers". What this means is inserting the "corner" pieces of the face with a solved cross along with the edge piece that goes along with it in the corrrect orientation too.
While solving a cross is done completely intuitively, there are several tips to solve the cross faster and more efficiently. Such as:
The way to solve F2L is also mainly done intuitively, but there are algorithms that can be learned instead to solve F2L. A PDF of those algorithms can be found here. They are shown with a picture of that particular case and their algorithm in Cube notation.
This video shows how to solve an efficient cross
This video shows how to solve F2L.