Every note has a pitch, determined by the fundamental frequency of the sound wave that produces it. When you have two different notes, you have two different pitches, caused by two different frequencies. The distance between those pitches is called an interval, and corresponds to the ratio of the note's frequencies. For example, if one note is an octave above a second note, their frequencies will have the ratio 2:1.
In Just Intonation, which was historically common in Western music at least into the Baroque period, the intervals are determined by forcing the notes to have frequencies that are small whole number ratios to each other. For example, a perfect fifth is 3:2, a perfect forth is 4:3, a major third is 5:4, and a minor third is 6:5. In historical western music, they usually stopped there (the interval 7:6 is not used), and all other intervals are then combinations of these intervals. The benefit of this system is that, because the frequencies are numerically related to one another, they sound very consonant, or pure to the ear. The downside is that those pure intervals only exist in one key; outside of that, the intervals start to break down.
For example, if you start at A♭, and go up a major third you get to C (with a ratio of 5:4 = 1.25). If you go up another major third, you reach E (5:4 x 5:4 = 25:16 = 1.5625). If you then go up one more major third, you reach G♯ (5:4 x 5:4 x 5:4 = 125:64 = 1.953125). As you can see, this doesn't quite reach the next highest A♭ (which must have a ratio of 2:1, by definition). The basic mathematical problem is that no matter how many times you repeat any of these ratios (other than the octave), you will never get back to a power of two, which means you will never get back to the starting pitch (in any octave). Which further means that you either have to have an infinite number of notes in your scale, or you have to add in an impure interval somewhere to get back to where you started.
If you want to play in lots of different keys and maintain these pure intervals, one possible solution is to use instruments that can freely vary their pitch, like violins or voices, so that you can always adjust to the correct just interval. This is often done subconsciously by singers, even in western music, just by finding what sounds good.
However, many instruments cannot do this; especially keyboards and fretted instruments, both of which have to have a predefined scale. One possibility for them is to start adding a bunch of extra notes: Make A♭ and G♯ be two different notes, for example. However, there could be an infinite number of such notes, so things can get pretty insane pretty quickly, like this keyboard that has 24 notes per octave:
This approach hasn't really been pursued much in western music.
Instead, the solution that Western music has devised is called 12-Tone Equal Temperament (12-TET). Rather than forcing frequency ratios to pure-sounding whole number ratios, it makes a compromise to divide each octave into twelve equally-sized intervals, and adds them together to get larger intervals. The frequency ratio required for this small interval (which happens to be a half step) is the 12th root of 2 (21/12 = 1.059463...), since multiplying it by itself 12 times gives you back 2 (an octave). In fact, the term "ratio" is a bit of a misnomer, since this is actually an irrational number.
If you combine 7 of these small intervals (27/12 = 1.4983...) you get a close approximation to a pure perfect fifth (3:2 = 1.50). If you combine four of them (24/12 = 1.2599...) you get a less-close approximation to a pure major third (5:4 = 1.250) which is still somewhat serviceable. But the key point is that by making all of the intervals equally out of tune, no key is any worse than the other. You do have to sacrifice the pureness of the intervals, but it is a relatively subtle effect to the untrained ear.
Here is a video that demonstrates both types of intervals:
TLDR: Just intonation refers to using mathematically simple ratios between a pair pitches; justly tuned intervals are objectively more consonant than non-justly tuned intervals. However, it is mathematically impossible to construct a scale of fixed pitches in which every possible interval is justly tuned, and some may even sound quite bad. 12 TET is one way of compromising when tuning a scale; it equally detunes all intervals, so that no single interval is either perfectly just, nor terribly far away from being just.