There is a lot of it about, using these three terms interchangeably. They are not interchangeable, They do not refer to nearly the same things. A solar system is a group of planets around a single star. A galaxy is a group of stars all huddled together, and the Universe is everything-that-exists. Each one is immensely bigger than the one before.
Our Solar-System
At the centre of our solar system is the sun. It is a fairly average star. Nine planets orbit it.
- Mercury — orbiting at 0.39Astronomical Units (AU)
- Venus — 0.723 AU
- Earth — 1 AU
- Mars — 1.524 AU
- Jupiter — 5.203 AU
- Saturn — 9.539 AU
- Uranus — 19.18 AU
- Neptune — 30.06 AU
- Pluto — 39.53 AU
Seen from the northern hemisphere, they all go round anti-clockwise. Everything in this solar system goes round the same way1. From the southern hemisphere of course, everything goes round clockwise, except the sundials there go anti-clockwise.
So, quick, what is the distance from Earth to Mars?
If you said 0.524AU, then you are wrong. Remember they are both going in circles around the sun. They could be anywhere on that circle. The answer is just as likely to be 2.524AU; both happen about once every two years. The rest of the time it is somewhere between the two.
As the orbits get larger, the planets are moving slower along their orbits and for that reason and because the orbits are larger, they take longer to get around.
The planets near the sun are made of rock, the ones further out are made of gas. It seems reasonable to assume (although I've not seen it proved anywhere) that this is because of a grading process when the solar system was formed. We may therefore assume that other solar systems are made similarly.
Planets can have moons. Generally moons are a lot smaller than the planets. (Our Moon is much larger than we might expect, and it is much smaller than Earth.) Larger planets seem to get more moons.
In addition to planets there are other things orbiting the sun: asteroids and comets. The asteroids are mostly between the orbits of Mars and Jupiter, all orbiting the sun just like the planets. The comets are much further out than Pluto. We only see them when they get knocked into very elliptical orbits that take them close to the sun. Asteroids are made of rock and comets are made of frozen gas.
Other Solar Systems
It seems likely that other stars were created in the same way ours was, and therefore they will have planets too. We don't know how similar to ours they will be, but it seems reasonable to suggest that they will share the major characteristics. The closer planets to the star will be closer together than those further out. The inner planets will be rock and the outer ones will be gas. Gas planets will be much bigger than rock planets. Big planets will have more moons.
See this Map of Nearby Stars.
Our Galaxy
Our galaxy is a flat spiral with a radius of 50,000 light years and contains around a hundred billion stars. We are about 30,000 light years from the centre. The Milky Way you see at night is the stars in the spiral arm looking outwards. Our view of the centre of the galaxy is blocked by interstellar dust. That's a pity because it would be a phenomenal sight. There's a central bulge that's about 16,000 light years diameter, but where we are the disc is only 1,000 light years thick. The galaxy is revolving, taking a little over two hundred million years for one revolution.
Where we are, stars are about four light years apart, give or take a bit. There are clusters of stars that are much closer together, but still a long way apart by solar-system standards. The Pleiades is a cluster about eight light years across containing about a thousand stars. That's an average separation of ten light months, or fifty thousand AU. In the centre of the galaxy there are likely to be bigger and closer clusters. There stars may only be light days apart.
Take a ball-point pen and make the smallest dot you can (about 1mm.) The outside edge of that dot is the orbit of Pluto — twelve billion kilometres across. Then make another dot twenty two feet away. That is the nearest star to us. Keep making dots at about the same separation until you have made a hundred billion of them and covered an area one hundred miles across. That's our galaxy.
Other Galaxies
Distances between galaxies gets a bit complex. It turns out there are two dwarf galaxies orbiting our galaxy so close that they will be torn apart fairly quickly. Other than those the Magelenic Clouds are a couple of hundred thousand lightyears away, and the Andromeda Galaxy is two and a half million light years away. Galaxies can be spiral like ours and Andromeda, eliptical like the Magelanic Clouds or irregular. Generally, only small galaxies are irregular.
The Universe
it's called Uni-verse because there is one of them. "Universes" is a contradiction in terms. There are reasons to refer to Universes, such as parallel universes in science fiction, but so far they are still in the realms of fiction. One is all that we have and it contains everything that exists. It's about thirty billion light years across and it contains about one hundred billion galaxies.
Let's have another 1mm dot, but have this one hanging in space. That dot is our galaxy. Make another one an inch away. That's the galaxy in Andromeda. Keep making dots until you've made another hundred billion of them in a sphere two hundred and eighty metres across. That's the Universe. Or try another one, on a smaller scale. Take a perspex sphere about five feet in diameter. Now blow cigarette smoke into the sphere until it's full of smoke in tendrils and filaments. That box is the Universe, and each individual smoke particle, just five thousandths of a millimetre across, is a galaxy. One hundred billion of them.