All,
May I try another explanation? It parallels Tobin's, but to my mind is more complete:

Any object in orbit around another experiences a force tending to pull it apart in a radial direction. This is because the whole object travels at the speed required to keep in orbit a point of the same mass at the position of its 'centre of gravity', or mass centre. A part of the object further out from the mass centre is travelling at the same speed but that is too fast for the wider orbit. If it was seperate, it would spiral out. An object, or part of the object further in at the same speed is going too slowly and will spiral in.

For a small artificial satellite the force is negligable, but for an object as big as the Earth it is large, and as the Earth is flexible, it bulges, towards AND away from the Moon. The rock bulges are small, but real and measureable, the water bulges are obvious in a large enough body of water. Sun tides are much smaller than Moon tides, because the Sun's gravity gradient is much less - it changes less across the diameter of the Earth.

This way of thinking about tides is more useful than 'gravity less further away' as it may be extended to other situations, for instance a skyhook or space elevator. I'm sure it is well known, but I came to it in a Pauline moment, on reading Larry Niven's story "Neutron Star". Which shows the educational power of Science Fiction!
John