Instead of an elliptic curve, imagine the curve of this river:

I can describe a position on the river between mile 0 and mile 7 by using both X and Y coordinates, much like a private key describes an X and Y coordinate on the elliptical curve.

You will see however that if I only tell you the X coordinate, you can still figure out where on the river you are. This would be the "compressed" river coordinate, telling it to someone would take half as much data.