As far as I know, a Cartesian plane is not a type of plane. It is just a way of setting up a coordinate system for a plane. A plane is any 2-dimensional flat surface that extends infinitely far in all directions. There are several types of coordinate systems for a plane, but the two most common are Cartesian coordinates (named for the French Renaissance mathematician/philosopher Rene Descartes), and polar coordinates.

Cartesian coordinates are based on taking two arbitrary straight lines in the plane that are perpendicular to each other, and an arbitrary unit of measure. Using the unit of measure, you can assign a number to each point on each of the two lines, namely its distance from the origin, which is the point where the two lines intersect. Points on one side of the origin are arbitrarily assigned positive numbers, points on the other side, negative. The two lines are called the x axis and the y axis. Now for any point in the plane, drop a perpendicular to the x axis. The number assigned to the point on that axis where the perpendicular line intersects it is the x coordinate of the original point (if the original point is already on the x axis, just take the number assigned to that point on the axis). Similarly for the y coordinate. Thus, every point in the plane can be identified by its x and y coordinates. Conversely, for every (ordered) pair of real numbers, there is exactly one point in the plane with those numbers as its x and y coordinates. For example, the coordinates (2,3) identify the point 2 units to the positive side of the y axis and 3 units to the positive side of the x axis. I said ordered pair to distinguish (2,3) from (3,2). The first number of the pair is, by convention, the x coordinate; the second, the y coordinate. By convention the x axis is drawn as a horizontal line with the positive side to the right of the origin. The y axis is a vertical line with the positive side pointing up. So (2,3) is 2 units to the right of the y axis and 3 units above the x axis. The coordinates of the origin are (0,0).

I won't describe polar coordinates in detail, because that is outside the scope of your question. But when you identify points on the surface of the Earth by latitude and longitude, you are using something very similar to polar coordinates, except that the surface is a sphere (approx.), not a plane, and both coordinates are angles. On a plane, one coordinate would be an angle and the other a distance.

Cartesian coordinates are based on taking two arbitrary straight lines in the plane that are perpendicular to each other, and an arbitrary unit of measure. Using the unit of measure, you can assign a number to each point on each of the two lines, namely its distance from the origin, which is the point where the two lines intersect. Points on one side of the origin are arbitrarily assigned positive numbers, points on the other side, negative. The two lines are called the x axis and the y axis. Now for any point in the plane, drop a perpendicular to the x axis. The number assigned to the point on that axis where the perpendicular line intersects it is the x coordinate of the original point (if the original point is already on the x axis, just take the number assigned to that point on the axis). Similarly for the y coordinate. Thus, every point in the plane can be identified by its x and y coordinates. Conversely, for every (ordered) pair of real numbers, there is exactly one point in the plane with those numbers as its x and y coordinates. For example, the coordinates (2,3) identify the point 2 units to the positive side of the y axis and 3 units to the positive side of the x axis. I said ordered pair to distinguish (2,3) from (3,2). The first number of the pair is, by convention, the x coordinate; the second, the y coordinate. By convention the x axis is drawn as a horizontal line with the positive side to the right of the origin. The y axis is a vertical line with the positive side pointing up. So (2,3) is 2 units to the right of the y axis and 3 units above the x axis. The coordinates of the origin are (0,0).

I won't describe polar coordinates in detail, because that is outside the scope of your question. But when you identify points on the surface of the Earth by latitude and longitude, you are using something very similar to polar coordinates, except that the surface is a sphere (approx.), not a plane, and both coordinates are angles. On a plane, one coordinate would be an angle and the other a distance.