The Sun is a pretty mundane star, like many, many others in the sky, although it is quite large — it's in the top 10% of stars by mass. However, it has one very special feature as far as we are concerned; and that is its closeness to Earth, some 250,000 times closer than the next-nearest star. Because of this, it appears large in the sky, compared to other stars; and its heat and light keep the Earth from freezing, and in fact, fuel all life on earth.
The Sun is huge: 1,390,000 km. across, compared to the Earth, which is a tiny 12,756 km. across, and is more than 320,000 times heavier than the Earth. In fact, the Sun contains more than 99.8% of the total mass of the Solar System!
To get an idea of the incredible scale of the Solar System, look at this diagram of the sizes of the planets (and Pluto), with a corner of the Sun in the background — at the same scale. Notice how the Earth could comfortably fit under one jet of gas from the Sun — many times over!
The Sun has a very dense core in which its energy is generated, at a temperature of 15 million degrees Celcius. The Sun is mostly (about 70%) made of hydrogen, but at the core, this is slowly being converted into helium by nuclear fusion; "slowly" meaning that 700 million tons of hydrogen are converted into 695 million tons of helium every second. The "lost" 5 million tons per second of mass is converted to pure energy, in the form of incredibly intense gamma rays. Fortunately for us, as this energy makes its way out through the layers of the Sun — taking thousands of years — it is continuously absorbed and re-emitted in less energetic forms, until it emerges mainly as heat and light.
Surrounding the bulk of the Sun's mass is a layer called the photosphere, which is only about 500 kilometres thick, but which is totally opaque; this is what we see as the "surface" of the Sun. The photosphere is relatively cool, at only 5,500 degrees Celcius. Cold dark spots (sunspots) can sometimes be seen in the photosphere; these are actually only cold in relative terms, at about 3,500 degrees. Sunspots can be as much as 50,000 km. across; but nobody really knows what causes them, or how they work.
Surrounding the photosphere is the chromosphere, a layer of gas about 4,000 kilometres thick, glowing with a distinctive red colour (although it's almost completely transparent, and so normally invisible).
Beyond the Chromosphere is the corona, a huge "atmosphere" of very thin but extremely hot gas, at about 2 million degrees Celcius, extending millions of kilometres into space. Nobody quite knows why it's so hot, when the Sun's surface is much cooler.
Extending into the hot corona are solar prominences, relatively cool (10,000 degrees or so) loops and tongues of gas suspended in the far hotter corona. Prominences can be as far as 50,000 km above the photosphere, and can be as much as 10,000 km thick and 600,000 km long. (You could fit Uranus, the third-largest planet, under one of those prominences!) Solar prominences can occur in active regions of the Sun (ie. where there is significant sunspot activity), or in quiet regions.
Prominences can last several months, and hang suspended, sometimes moving up or down, in the corona; then, at the end of their lives, they typically shoot upwards at several hundred km/s, and dissipate. Since prominences are denser, and hence heavier, than the surrounding coronal gas, there must be something holding them up; although the explanation for this is not fully understood, it is almost certainly to do with the complex magnetic fields of the Sun.
As you can see in the picture here, these phenomena are extraordinary and beautiful events. So, how can one observe the chromosphere, corona, and (when the Sun is particularly active) prominences, and solar flares? The immediate problem is that they are too dim to be seen next to the Sun's disk, since the light from the photosphere would totally flood them out — the corona, for example, is a million times fainter than the photosphere.
So, if we block out the photosphere, the Sun's fainter features should become visible — in theory. This works in space, but on Earth it's no good — the remaining light from the sky (Sunlight scattered by the atmosphere) means that only blue sky would be visible. So us earthbound mortals can never see the beauty of the Sun's structure — unless we can arrange for the Sun's photosphere to be blocked by something outside the atmosphere. Wouldn't that be cool! So can the Moon do it?
Well, really, we are asking a lot. In order for the chromosphere, corona and solar prominences to become visible, we need the photosphere to be totally blocked out; that means that the Moon needs to be big and/or near enough to cover it completely. But, if the Moon is too big and near, it will also cover up the features we want to see! So we need it to be just the right size and distance from the Earth.
And we're in luck! By an amazing — and very fortunate — coincidence, the Moon is just exactly the right size and distance from the Earth to do the job. In fact, if it was just a little smaller or farther away, we would never see a total eclipse.
Actually, as explained in Mechanics of Solar Eclipses, because the Moon's orbit is slightly elliptical, the degree to which the Sun is covered during an eclipse varies slightly. In fact, on average, the Moon is 0.531° in angular diameter, which is smaller than the Sun's average of 0.533°; though the Moon at perigee can be up to 0.568°, which easily covers the Sun. This is why more central solar eclipses are annular than total. The statistics page has more information on the sizes of the Sun and Moon.
All of this means that during a total solar eclipse, and only then, the Sun's corona can be seen from Earth; and this is what makes an eclipse such a rare and wonderful event. It is made even more rare by the fact that a total eclipse can only be seen from a tiny part of the Earth's surface; so you have to make sure that you're in the right place at the right time!
By the way, the Sun is about 5 billion years old. It has used up about half of its hydrogen fuel, and should keep going for about another 5 billion years or so. After that, watch out!