You know what points are on a plane when the point's (x,y,z) values are plugged into the plane's equation and the result is an equality. For example, if you have plane z = 3x+4y, you know that point (1,2,11) is on the plane because when you plug the points into the equation you get (11) = 3(1)+4(2) = 3+8 = 11; thus this is an equality so the points are on the plane.

You know that planes intersect whenever they are not parallel.