in a nutshell texture mapping is when you map a point in 3D space to 2D space. You should hopefully be familiar with 2D space from school. 'Mapping between two points' should be pretty self explanatory, but if it isn't then in this context it means that you take a point in 3D and put it to 2D. The vertex will have the colour of the texture at that point in 2D. If two points on a mesh are mapped to the same point in 2D space then they will have the same colour.
you might be able to figure out why you "cut the mesh" from that. You don't generally want UV's of the same mesh to 'overlap' with each other. Take a look at some of the texturing videos here, you'll start to get the idea behind it
that's a "Ch" pronounced as a "K"
Computer skills I should have:
Maya, XSI, Photoshop, AfterEffects, Motion, Illustrator, Flash, Swift3D