Relative pitch is the ability to hear two notes and determine their relationship.
For the purpose of simplification, I'm going to split the concept of relative pitch into two sub-concepts:
Basic relative pitch: you can hear two different notes, and you know straight away which note is the higher note and which note is the lower note.
Advanced relative pitch: you can hear two different notes, and you know straight away which note is the higher note and which note is the lower note AND the exact distance between these two notes.
Starting off with basic relative pitch, which is just listening to two notes and working out which one is higher and which is lower - you can practice this on any instrument just by moving up and down scales and listening to the changes in sound. People who are "tone deaf" are those who can't hear the differences - they hear a sound and don't know the pitch relationship between it and any other sound.
Once you've got identifying which sound is higher to the point where you can get it right every time without fail, then it's time for relative pitch interval training. This is where it gets trickier.
Here's all the musical intervals in pop music within a range of one octave:
There's many different ways to learn what the distances between these intervals sound like. Here's a common type of method that is taught - associating an emotional feeling or qualitative aspect with each sound (forgive the typos and incorrections, the following chart wasn't created by me and I've left it as-is):
The problem with this method is that not everybody hears these intervals in exactly the same way, especially once you add cultural aspects and different people's musical upbringing into the mix. Someone who listens to lots of heavy metal or punk won't find a fifth to sound "empty" or a minor second to sound "irritating". A blues afficionado probably won't find a tritone all that harsh, either.
I've always found that the best way to learn relative pitch intervals is "relatively" - to listen to examples in popular music and memorise them, and then when they come up in other songs they will trigger your memory because you will be able to link them back to the songs you already are very familiar with. The more familiar you are with the songs, the better this works. For instance, a major seventh is a very hard interval to recognise straight off the bat, because the two notes are very far apart. However, it's easy as piss if you remember that the major seventh interval forms a key component of the theme from the movie "Superman", which is an interval that nearly everybody knows subconsciously because everybody went and saw a Superman film at one point in their lifetime. The interval of a major seventh happens at 0:51 where it goes really high:
Now, if you happen to hear two notes anywhere else, all you need to do is sing "Superman" to yourself while you're listening (not necessarily out loud, just in your own head will do the job) and if the notes match up, you know you're dealing with a major seventh. If not, then it's some other interval.
If you've built up a list of tunes to memorise for all the different intervals, then you can just keep trying different intervals in your head until you get the correct one. Using this system of matching intervals to well known music, you can learn to recognise any interval when you hear it, but instead of using movie soundtracks, I'm going to give you k-pop singing examples of each interval that you can memorise. Most of the following examples have the low note as the starting note and the higher note as the second note in the sequence. Listen for the pitch change.
Starting with unison - in other words, no pitch change. I don't have to give an example for that, do I? Good. Moving on.
The minor second: Krystal's opening line in f(x)'s "Rum Pum Pum Pum" at 0:09:
The next interval on the list, the major second interval is the first two notes Minzy sings in 2NE1's "Falling In Love" at 0:27. Listen to this, then listen to 4Minute's bit and notice how the distance in 2NE1's song is wider.
It's also in Girls' Generation's "Paparazzi" at 2:10, listen close as it's hard to tell apart from the minor second, but listen to this back to back with the f(x) example and you'll eventually get used to the difference:
The minor third interval is common in k-pop. The first ten seconds of TVXQ's "Mirotic" vocals from 0:23 just works one solitary minor third interval over and over.
Also the entire vocal line of BigBang's "Monster" from 1:03 to 1:13 is ALL minor thirds. If this doesn't get you used to the sound of it, nothing will.
The major third is also common, here it is in BEG's "Abracadabra" verse singing line at 0:48:
A very clear perfect fourth is in the "kiss me baby" line of the chorus of T-ara's "Day By Day" at 1:38:
The same interval starts off the verse melody of Day By Day at 1:06.
Everything Sunny sings in Girls' Generation's "I Got A Boy" from 1:33 to 1:37 is also working a perfect fourth interval:
Advanced examples: the chorus line after "Hey Boy" in Hyuna's "Bubble Pop" at 0:47 goes up and down the major scale, from the starting note to a major second, major third, perfect fourth, and then back down to the start again. T-ara's Day By Day chorus does a similar climbing movement (from the word "baby") but it climbs up the minor scale instead, to a major second, minor third and perfect fourth, and doesn't come back down quite so directly.
The tritone or flat 5th is quite an unusual-sounding interval that is rare outside of heavy metal so in the smooth style of k-pop it's extraordinarily unusual and most songwriters go to great lengths to avoid using it ever. However, the tritone does also have a functional use in blues music, and so it also appears in blues-inspired k-pop such as Lee Hi's "1, 2, 3, 4" where she jumps up to it briefly at 0:38 with quite a raspy voice:
The chorus to Lim Kim's "All Right" that begins at 0:08 is mostly just a sequence of descending perfect fifths.
Also a perfect fifth - the "oh oh" bit in Super Junior's "Sorry Sorry" at 0:32:
The minor sixth is rarely used, but here it is in The Seeya's "Be With You" at 1:17:
Major sixth: The first notes in the chorus of F-ve Dolls + Dani's "Can You Love Me" at 0:51:
Also major sixth - the jump-ups in the slow section of Ailee's "I'll Show You" at 0:23 to 0:25:
Also, every Davichi song ever.
Rare in k-pop, a minor seventh can still be found - it's in the first chorus notes of F-ve Dolls' "You Cheated" at 0:55. Actually the note starts from the starting point, moving up to a minor seventh and then steps down, resolving on a major sixth relative to the first note, an example of the type of sexy suspended melodic movement that is all over the F-ve Dolls mini-album and helped make it my favourite mini of 2013. Just on the off-chance that you actually give a fuck about that kind of thing.
The major seventh is as rare as hen's teeth and in the small time I had to put this together I couldn't find a single k-pop example of it, but I'm sure there's one out there.
EDIT: thanks to Sera S for submitting Younha's "Run"! The relevant note is at 0:46, and the melody line goes from the starting note to a major second, then back down to the starting note and then up to the high major seventh.
An octave is being worked back and forth in "Something" by Girl's Day from 0:54 to 1:00.
There's no point going beyond an octave as the musical scale wraps around at that point so you use the same cues to identify a ninth that you would use for a second.
Here's a summary in a chart if you need a quick reference:
Memorise how those parts in those songs sound, and now you've got a working "pitch memory" that you can use in any situation where you might need to recognise pitch intervals, such as learning how to sing or play a song that you're not familiar with, or knowing if you've hit a certain interval correctly.
This is not a post about which singer is better. Do NOT take this post over to your favourite vocal thread and go "look, my bias can hit interval x, she's better than your bias". Any singer can hit any interval if they learn how (especially if they use this method I've outlined), even if their voice sucks. I can hit ALL of these intervals, and my own singing voice is fucking shit. This post is about you learning to identify pitches, it's not about proving who is better than who or for others to identify what your bias is or how mentally ill you are.
I hope this has been helpful to those of you learning music, and not too annoying for the rest of you!