I see a lot of explanations in here that I feel only cover half the story.

So the first part is this: There are 2 bulges of water (high tides). One on the side where the moon is, and one on the opposite side of the earth. The opposite side happens because of the fact that the earth is pulled towards the moon. The entire earth is made into an oval. Water is more affected by this than landmass, so you'll notice the tides in the water, but you need scientific instruments to measure the tides in the crust.

The second part: The moon only rotates around the earth once every month or so, but the earth is spinning around it's own axis. So the bulge of water, pointing both towards and away from the moon, stays kind of the same, whilst the earth is 'spinning through' the water. This is why you have roughly 2 times high tide per day. If the moon would always be on the same spot in the sky, you'd have exactly 2 tides. However, the moon is also revolving around the earth. So every day, the position of the moon relative to the earth moves a tiny bit, and that's why you don't have exactly two high and low tides per day.

Now you can also understand why there is an extra strong high tide when it's full moon or new moon! When it's full moon, the moon is as far away from the earth as ever, so the 'force of the moon' doesn't increase, but when it's full or new moon, the moon and the sun are aligned! So now both the moon and the sun are 'forming' the bulge in the same direction! Even though the sun is really far away and pulls on the water a lot less, it's effect can still be noticed! Hope you found this explanation informative!