Getting a signal from A to B
Radio waves are just low frequency light (and conversely light is just high-frequency radio waves). They're made of exactly the same stuff. This means that radio waves don't bend. How often are you able to see round a corner, after all?
This means that if you can see the person you're communicating with, things are simple. Because radio waves are lower frequency than light, they also go through things to some extent - people, houses, trees, etc, so depending on the frequency, it'll also get a little further than what you can see.
But in order to communicate with someone over the horizon that you can't see, we need some method of getting the radio waves round the curve of the Earth.
There are a number of ways of doing this
Build a piece of equipment that listens on one frequency and simultaneously transmits on another, and put it on a high hill within site of both stations. This is a repeater, but it only really extends the range to maybe 100 miles. This is due to limits of the height of the hill.
To get your repeater much higher, you can shoot it into space on a rocket and call it a satellite. In Low Earth Orbit, it will be 1200 or so miles up, and this will increase the maximum range between two stations to maybe 3000 miles. Obviously, building and launching sensitive equipment into space is quite hard. It is rocket science after all. Also, satellites in Low Earth Orbit have to travel very quickly to stay in orbit at the height they are, so they are only overhead for 10 minutes at a time. LEO satellites travel at 5 miles a second, and this can mean that you have to adjust for the Doppler effect (even though radio waves travel at the speed of light).
If the satellite was put about 22300 miles, (36000 kms) away, it would only need to travel more slowly, and could stay above the same point of the Earth all the time. This would be a geostationary satellite, and would be able to cover a lot more of the world, increasing the possible distance contacts could be made from. Because the satellite wouldn't appear to move in the sky, it wouldn't be affected by the Doppler effect, and the satellite would be available all the time, rather than just for brief windows as it passed over your patch of sky. This is what satellite TV satellites are.
The downside of a geostationary satellite is that it's a lot further away, and so signals are weaker.
The higher the frequency of the radio wave, the more it acts like light. Light doesn't go through most things (apart from windows, air, etc), and is blocked by physical items. Trees, houses, people. However, mirrors allow light to be bounced off hard objects, and thus around corners. Wouldn't it be good to be able to bounce radio waves around corners (or the curve of the Earth). For this, we need some mirrors in the sky.
Aircraft are hard metallic objects quite high up. Radio amateurs have known for quite a while that if you use high enough frequency radio waves, you can aim your transmissions at an aeroplane, and they will reflect the signal further away. Aeroplanes are generally only up to 5 miles high though so it doesn't extend your range a lot. They also move, and aren't always in the right place in the sky at the time you want them to be.
Meteors hit the Earth's atmosphere hundreds of times a day. Every time a meteor burns up in the atmosphere, it produces a little patch of intense ionisation, which is very reflective to higher frequency radio waves. These patches of ionisation are essentially random in time and location, and are very short lived - 0 to 2 seconds. Although amateur radio operators can't know in advance where and when a meteor will burn up in the atmosphere, the statistics say it is likely to happen at least once every thirty seconds. They have thus developed a method of communicating via data, where they transmit small amounts of data repetitively for 30 seconds. The chance is high that a meteor will have burnt up in the atmosphere during that time and reflected the signal. East to West stations transmit between 0 and 30 seconds in the minute, and West to East stations transmit between 30 and the top of the minute.
The higher the frequency of a radio wave, the more likely it is to bounce off something.
Very high frequency radio waves can bounce (or rather be scattered) off rain, and to a lesser effect snow and ice pellets. This is exactly how rain radar works, where they measure the signal that bounces back when a signal is transmitted in a specific direction. However the scattering also reflects forwards as well, making it possible to contact stations further away.
The moon can also be used to bounce radio waves off. This is one of the hardest achievements to unlock in the game of amateur radio. The moon's surface is quite reflective to light (you can see it after all), but it's not very flat, and is covered in craters and dust, and it's not very reflective to radio waves. The distance from Earth to the moon means that most of the signal transmitted is lost in the journey. A small amount of what's left is reflected back, and most of that reflected signal is lost on the way back to Earth. The signal received back on Earth is a miniscule proportion of what was transmitted. This means that high power amplifiers and high-gain antennas are needed on the transmitting side, and high-gain antennas and pre-amps are needed by the receiver. Even so, the signal is often too weak to be able to conduct voice conversations, and so digital modes or Morse code is often used. Modern weak-signal modes and computer technology mean that it is easier nowadays, but still not easy. Using the moon as a reflector means that you can communicated between two stations wherever the moon is above the horizon, giving you total global coverage (depending on where the moon is). Signals bounced off the moon take about 2.5 seconds to return to earth, even travelling at the speed of light.
Wouldn't it be nice though, if there was something that was pretty high up, and available pretty much all of the time?
Unlike all the methods above which need high frequency radio waves (that act more like light), the ionosphere can also be used for bouncing radio signals off. The Sun transmits billions of watts on all frequencies in all directions. Some of this power ionises the Earth's ionosphere, and this can be used for bouncing lower frequency radio waves from. This method of bouncing is one of the most reliable, although it does depend on solar activity - sunspots, flares, coronal mass ejections. A radio signal can travel 3000 miles with a single bounce from the ionosphere, but often the reflected signal bounces back down to Earth and back up again, possibly multiple times, giving total global coverage. If you need fairly reliable global communications, without the need for any shared infrastructure such as underseas internet cables or satellites, this is probably your best bet.
As you can see, there are numerous different ways you can get radio signals further than you can see, each with different pros and cons. There is more information about radio propagation on Wikipedia.
Wikipedia also has more information about amateur radio satellites.