Tidal forces
This simulation shows tidal forces at points around the equator (viewed from above the North Pole) and a view of the Moon from the Earth (to show how the phases of the Moon correllate with the tides).
It illustrates how the tidal forces of the Moon and Sun combine to produce spring tides, and oppose to produce neap tides. In between these extremes the tidal influence of the Sun causes "priming" or "lagging" of the tide away from the stronger influence of the Moon.

The Earth and Moon are shown at 15 times actual linear size so that the tidal forces around the Earth and the phases of the Moon can easily be seen.
One side effect of this (in the view of the Moon from Earth) is that you see Solar eclipses rather more often than you would in real life!
The view of the Moon from Earth has a few other peculiarities: The view is as it would be from the centre of the Earth - but without the Earth (or clouds or daytime blue sky) blocking the view, and "Up" is the same direction as it would be for someone standing on the North Pole. A view from the equator would be rotated through 90 degrees.
The blue line in the main view shows where the Earth has just passed in its orbit around the Sun.

What causes tidal forces?
The tidal forces shown in this simulation are simply the difference in the gravitational force (from the Moon or Sun) felt by a  mass at some point on the Earth's surface compared to the force felt were the mass at the centre of the Earth.
An object on the side of the Earth facing the moon experiences a slightly stronger attraction than the Earth as a whole. This gives a tidal force towards the Moon - and away from the centre of the Earth.
An object on the side of the Earth facing directly away from the moon experiences a slightly weaker attraction than the Earth as a whole. This gives a tidal force away from the Moon - and also away from the centre of the Earth.
Objects on the surface of the Earth, the same distance from the Moon as an object at the centre of the Earth, feel the same strength of attraction - but in a slightly different direction. This difference gives a tidal force towards the centre of the Earth.
So both the stretching and squashing of the Moon's tidal forces act to try to produce two lumps on opposite sides the Earth - one facing the Moon, the other facing away.
The Sun also produces tidal forces on the Earth, but these are smaller than the Moon's because although the Sun's pull on the Earth is much greater, it is so much further away than the Moon that the difference in its pull on the nearer and further sides the Earth is actually less.
The tides result from the sum of both the Moon's and the Sun's tidal forces.

How strong are tidal forces?
Tidal forces on the Earth from the Moon are 0.033 times less than the attraction between them.
Tidal forces on the Earth from the Sun are 0.000085 less than the attraction between them - because the Sun is much further away.
The maximum tidal pull is about 0.000001 N/Kg
In comparison the Earth's gravitational pull at its surface is about 10 N/Kg;
10 million times stronger.

If tidal forces are so weak compared to the Earth's gravity - why are the tides noticable?
Because the tidal forces at a particular point on the Earth surface change as the Earth rotates - so that (albeit small) tidal lumps move over the surface of the Earth.
The existance of large oceans of water free to slop around in response to tidal forces also contributes to the height of ocean tides - as water can flow into the tidal lump from a much greater area.

Doesn't the body of the Earth itself move in response to tidal forces?
It does - but not as much as the liquid oceans.
If the land surface and sea bed rose and fell more freely in response to tidal forces, there would be less impetus for ocean tides to flow - as the sea bed would rise to fill more of the space that the tide now flows into (and fall to free more of the space where the tide would be ebbing).

Have the tides always been as they are now?
The rotation of the Earth (and the friction of water moving over the sea bed) carries the tidal lump ahead of the tidal force. The Moon's gravitational pull on this slightly offset tidal lump slows the rotation of the Earth; the energy and angular momentum being tranfered to the Moon which moves higher in its orbit.
The current rate of slow down in the daily rotation time is about 1.5 milliseconds per century.
This has a noticable effect over millions of years - try working it out for a particular time.
As the speed of the Earth rotation slows the speed of tidal flows will diminish as water won't need to move so fast to stay in the tidal humps (and the lumps themselves get smaller as the Moon's orbit gets higher). The dynamic effects of tidal flows - extreme high and low tides around certain coasts, tidal bores - will thus diminish as the tide becomes a more gentle affair.
This will also lead (as it has done in the past) to the rate of slow down decreasing; with the slower rotating Earth and more distant Moon givng less tidal drag.

Tidal effects around the Solar system
The rotation of the Moon has been slowed down by the friction of tidal movements (within the Moon) so that it now rotates at the same rate that it orbits the Earth (the same side of the Moon always faces the Earth) so that its tidal "lumps" stay in the same position.

Pluto and its moon Charon have a matched orbital and rotation period - with the same part of each facing the other.
The same will eventually happen to the Earth - if the Solar system still exists by then - with a day lengthening to a lunar month.

Mercury and Sun 3:2 resonance.

The frictional heat generated by tidal movements in Jupiter's moon Io gives rise to extensive volcanic eruptions.

Other notes...
The lunar tidal force on the side of the Earth facing the Moon is 6.3% stronger than that facing away from the Moon.
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As the Moon gets nearer to or further away from the Earth as it moves in its elliptical orbit, it's apparent size as viewed from Earth changes noticably. (The main view in this simulation - along the Earth's axis - is in a significantly different plane to the Moon's orbit, so doesn't give a good impression of the distance of the Moon from the Earth.)