Digimodes

The purpose of radio is to getting information from A to B. What that information is, is dependent on the user. It could be cat gifs, or recipes, or secret messages to spies. It could be a video streaming to your mobile phone.

When radio was first invented, the simplest way of getting information was to switch the transmitted signal on and off. This became Morse code.
Eventually, people worked out how to make the signal strength vary with the loudness of someone's voice (Amplitude Modulation). After that, they made the frequency of the transmitted signal vary with the voice (Frequency Modulation).

A very large amount of contacts made on amateur radio these days are done digitally using digimodes.
This is great if you don't like actually speaking to people!
The principle is the same as an old telephone modem. You convert your text/picture/video/voice into ones and zeros, and you then turn those ones and zeroes into sounds which are then played over the phoneline/radio waves.
A receiver "hears" the sounds, and converts them (hopefully) back into the same ones and zeroes that the transmitter sent.
One of the earliest digimodes invented was literally two alternating tones representing the ones and zeroes - RTTY. This is a Frequency-shift keying mode.
People realised that if you sent multiple tones at the same time, you could send your data faster - these were the Multiple Frequency-shift Keying modes.
Finally, people realised that you could use a single tone, and change the phase of the signal. These are called Phase-shift keying modes.

Difficulties

There are a few key differences between a telephone modem and an amateur radio digimode though.

All of the above mean that sending data over a radio link is not easy. You might get signal fading, or lightning causing crackles or an Over The Horizon radar (such as the Duga-1 near Chernobyl) interfering with your transmission.
It also is a lot slower than the telephone modems (which weren't exactly fast anyway).

Errors

Because it is likely that your signal won't arrive perfectly on the amateur bands, lots of digimodes make the assumption that errors will occur, and add in some extra redundancy so that whenever the inevitable errors occur, they can be detected, and corrected automatically.
Because the error correction is sent in advance of knowing if there actually was an error, it's called Forward Error Correction. It is mainly used in situations where you can't ask the sender to resend information if there was an error, such as on CDs, or on satellite TV.
A very basic example of this would be to send the message three times. If you received PutAHorse!BuyAHorse!BuyAHouse!, you could work out that the intended message was most likely BuyAHorse!
However, this is highly inefficient. Each transmitted message is three times as long as the actual message, which over low-bandwidth, globally shared resources is not the cleverest way of doing things.
Clever people have come up with much better Forward Error Correction systems than this. Hamming7,4, and Reed-Solomon are a couple of the most well-known ones.

Weak-signals

Where digimodes shine is when the signal is weak.
Computers nowadays can pull signals out of the background noise that a human can't hear, and recover the message from them.
This is really great in these days when there is lots of interference in cities, people might not have enough space to put up a decent antenna, or their licence might not allow them to use much power. And there's the one problem that we can't do anything about - low Solar Flux Index levels.
Even these compromised setups can still make their signal heard around the world. I have transmitted a 5W signal that was heard in Australia. Other people have done the same with much less power than that. That's some pretty impressive technology and software.
A general rule is, the lower the signal, the more slowly you have to send it.

There are two kinds of digimodes - automated, and interactive.