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Say we're on the piano and we're playing a melody in the right hand. I've noticed sometimes pianists will add another note to that melody that harmonizes with it and makes the melody sound fuller.

So basically two notes are being played at once on every note in the melody. The rightmost finger will play the actual melody and the thumb (on the right hand) will harmonize with that note.

Is there a name for this? I know it's not a chord. Is it just called harmony or something, or is there an official name for it?

Also, is there a certain interval that this is usually done? I assume the note needs to be in the scale so maybe it's not a static interval.

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If it's just one other note, then you can really just say they're playing an interval. Or just calling it a harmony is fine.

You are correct -- a chord consists of at least three notes. Pianists can also hit chords as they're playing a melody (which is helpful especially if they need to comp themselves).

The melody probably sounds fuller when a piano harmonizes with itself because now you hear multiple voices and multiple colors (the colors coming from the fact that there is harmony now).

If you are in-fact talking about improvisation as the tags show, then there is no rule for when a pianist can't play harmonies. It is up to them as the musician and the improviser. During their improvisation, if they feel they want a certain harmonic color to be thrown in, then that's when they would play an interval/harmony.

The note doesn't necessarily need to be in any sort of scale. Sure, if you're given a chord, it's usually nicer on the ear if in the beginning, you focus on sticking to the harmony of the chord. But again, it's completely up the pianist: if they want a harmonic color that's not within the chord or scale, then it's up to them what note to play. In fact, playing harmonies that aren't related to the scale can help define the climax of a improvised solo.

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Accompanying with a single note is rather common in almost all genres of (western) music. The richest sounds come from thirds and sixths; however a long string of either sounds a bit monotonous. Alternating thirds and fifths with a variable number of each allows one to a couple of choices of implied harmony on each note. A few fifths or fourths are possible but these seem a bit empty often and consecutive fifths sounds like one voice drops out. I prefer to use a string of thirds or sixths or rests or a few three voice spots. This allows for a bit of variety with the rests and chords used to emphasize certain notes.

To get started, try accompanying with lower thirds and sixths; you will quickly learn which ones sound best. It's really just two-part counterpoint, essentially first species. Note that one may form an interval of a third or sixth that is quite dissonant with respect to the underlying harmony; this of course adds a lot of emphasis.

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The usual name for this is double stops. It's a common expression among players of string instruments because it's a difficult and yet comparatively common effect. For pianists, doubling a melody is way easier, which is probably why they rarely bother naming the practice.

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The supporting harmony notes can be added in various ways. Sometimes the melody is doubled in thirds, sixths, or octaves where the added part moves in parallel motion. You could refer to this as the right hand playing in thirds or in octaves, etc. These are informal terms (except parallel motion, that's standard music theory lingo) but I think they are commonly used and understood. Of course, the supporting harmony doesn't have to use these intervals - others could be used - and it doesn't need to be parallel motion. But such doubling is common.

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Usually this is called layering. It's used when a hand is playing multiple notes at the same time, and they are different durations. For example, in the key of C Major, a pianist could be playing the melody with their right hand while holding a suspended C with their pinky. This is used to create harmony in higher notes without using the left hand.

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Harmony can be parallel, contrary or oblique (where the melody moves up/down and the harmony stays on one note). As a general practice, it's nice to mix them up a bit. Parallel would be the most common in RH piano harmony because you can run out of finger stretch pretty quickly with contrary and oblique.

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It's called filling in the harmony in the RH. There are several techniques. It may be a structured second voice, as if an additional singer had joined the group. Typically it will run a mixture of 3rds or 6ths below the melody, or be static. Or it may not follow the shorter notes of the melody, just filling in chords at certain points.

Here's a short example of what a real-life pianist might play to accompany a melody.

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