An interval is the distance between two pitches. When you sing "Happy Birthday," the first two notes ("Hap-py") are a major second apart. When you sing the Star Wars theme, the opening leap is a perfect fifth. Every melody you've ever heard is just a sequence of intervals. When you can identify those intervals by ear - and then reproduce them with your voice - you can learn songs faster, sing harmonies more easily, and improvise with confidence.
There are twelve intervals within an octave, but you don't need to learn all twelve at once. Start with these five, which cover about 80% of what you'll encounter in popular music: major second (two half steps - think "Happy Birthday"), major third (four half steps - think "Oh When the Saints"), perfect fourth (five half steps - think "Here Comes the Bride"), perfect fifth (seven half steps - think "Star Wars"), and octave (twelve half steps - think "Somewhere Over the Rainbow").
The "song association" method is the fastest way to internalize intervals. For each interval, memorize the first two notes of a well-known song that starts with that interval. When you hear an unknown interval, mentally scan your reference songs until one matches. Over time, you won't need the reference songs anymore - the intervals will have their own distinct "flavor" that you recognize immediately.
But recognition alone isn't enough for singers. You need to be able to produce intervals, not just identify them. This means hearing a note and then singing another note a specific interval above or below it. The training for this is straightforward but requires daily repetition: play a random note, then sing a specific interval above it. Check with the piano. Were you accurate? Adjust and try again.
Here's a concrete exercise that takes five minutes and trains both recognition and production: play a note on a keyboard or app. Sing that note. Now sing a major third above it (think "Oh When the Saints" from that starting note). Check with the piano. Was your third accurate? Now sing a perfect fifth above the original note. Check again. Do this with five different starting notes. Tomorrow, use the same five intervals but add minor third and perfect fourth. Build up gradually until you can produce all twelve intervals from any starting note.
The intervals that trip singers up the most are minor second (one half step) and major seventh (eleven half steps). The minor second sounds tense and dissonant - most people avoid it instinctively when singing. The major seventh sounds "almost but not quite" an octave, which confuses the ear. These intervals are rare in simple melodies, but they appear frequently in jazz, gospel, and complex pop harmony. You don't need to master them first, but be aware they exist.
Interval training is one of those skills where the first week is the hardest and each subsequent week gets dramatically easier. Your brain is building new auditory categories - literally creating neural pathways that didn't exist before. The first few days, everything sounds the same. By the end of the first month, intervals pop out at you when you listen to any music. It's like learning a new language: painful at first, then increasingly automatic.
For singers specifically, the most important interval to master is the third, because thirds are the basis of harmony singing. If you can reliably sing a major third or minor third above any note, you can harmonize with any melody in any key. That one interval is the gateway to the entire world of vocal harmony, and most singers never explicitly drill it.