Solar and lunar eclipses both happen when the Sun, the Earth and the Moon happen to be on the same line in space. As the Moon orbits the earth once every month, one might expect this alignment to happen twice every month, once when the Moon is towards the Sun (new moon) and again when it's on the opposite side (full moon). However, the orbit of the Moon is also tilted slightly, so the alignment can only happen when the Earth-Moon system is in a suitable position in its orbit around the Sun.
During the new moon, when the Moon is in the direction of the Sun, a solar eclipse can happen, as the Moon moves in front of the Sun, as seen from the Earth.
It is a coincidence that the Moon happens to be almost exactly the same size as the Sun in the sky. The Sun is of course much, much larger, but it is also very far away compared to the Moon, making it look smaller.
The Moon is rather small compared to the Earth, so its shadow is also not very wide. Because of this, total solar eclipses can only be seen in very small areas on the Earth. The last total eclipse of the sun in Finland happened in 1991. The next one will happen in the year 2126. Partial eclipses can be seen in larger areas.
The distance between the Earth and the Moon varies slightly, making the apparent size of the Moon change. Because of this, eclipses may sometimes be annular instead of total. In this case the Moon is perfectly in front of the Sun, like in a total eclipse, but it is not large enough to cover it completely, and the surface of the Sun can be seen as a ring.
See the wikipedia article for details.
A lunar eclipse happens during the full Moon. This means that the Moon and the Sun are in exactly opposite directions in the sky, with the Earth between them.
The Earth is large compared to the Moon, and its shadow is also wide enough to cover the Moon entirely. Therefore, a lunar eclipse can be seen everywhere on the night side of the Earth, and also lasts much longer than a solar eclipse.
The Earth's shadow consists of the penumbra, or partial shadow, and the umbra, or full shadow. The Moon moves first into the penumbra, getting slowly darker and redder, until it reaches the umbra and is entirely in shadow. Even in the full shadow the Moon can be seen as dark red in the sky. The red colour is due to scattered light from the Earth's atmosphere.
See the wikipedia article for details.
The tides can be seen on the coasts of large seas as a raising and lowering of the sea, repeating twice daily. The tides are caused by the Moon's gravitational force acting on the waters of the Earth. The daily variation is due to the Earth's rotation.
Tides also have a smaller monthly variation, due to the position of the Moon relative to the Sun. The Sun's contribution is not as significant because it's farther away. Near the new moon, when the Moon and the Sun are in the same direction, their gravitation forces add up and cause higher "spring tides". Near the full moon, the Moon and the Sun are in opposite directions, and the Sun's effect makes the tides lower.
The tides also affect the solid Earth, slightly deforming it. The Earth's surface move up and down by about 30 centimetres every day due to the Moon's tidal pull.
Tidal forces between the Earth and the Moon also work to transfer energy from the Earth's rotation to the Moon's orbital motion. This causes the Moon to spin up slowly and move away from the Earth at a rate of about 4 centimeters per year, and the Earth to spin down, making the days longer by about 15 microseconds per year. The process is very slow, but hundreds of millions of years ago the Moon was significantly closer and the days on Earth were shorter.
The process will end when the days on Earth are the same length as months. Then, the tidal forces are stable and no tides will happen on Earth. But in practice this will not happen during the expected lifetime of the Sun.