Tendency tones are tones that "tend" to move in a particular direction. This is essentially a statistical phenomenon that is style dependent and experience dependent. Two people that spend a lot of time with a particular style of music will likely develop a similar sense of what is common or what "tends" to happen in that style, but two people who focus on very different genres will likely develop a different set of expectations of what tends to happen. The brain regularly captures patterns and usually only brings things to our attention when an expectation has been violated given the patterns it was expecting. If everything goes as expected, the brain doesn't bother letting you know about it, like when you notice that the sound of a fan or air conditioner turned off, even though you didn't consciously realize it was on in the first place. The brain expected the noise to continue indefinetly so it didn't bring it to your attention, so when something unexpected happened—the noise stopping—it does bring it to your attention because of the violated expectation.
Musical tendency tones are very similar. We can distinguish between when a tendency tone moves (or resolves) the way it has done many other times in our experience, and when it hasn't. When it does resolve like we expected, that is the "normal" resolution, and when it doesn't, it's usually either an elided resolution, or it's still a normal resolution but is "implied."
In European classical music of the "common-practice period" (approximately 1600-1900) and also in "Tonal" music more broadly, the tendency tones are the leading tone, chordal seventh, and chromatically altered pitches. The descriptions on this page are describing specific tendency tones within this style of music.
The leading tone is the name of the seventh scale degree of the major scale (Ti). It is also found in minor scales when the diatonic seventh scale degree (the subtonic, or Te) is raised (Ti). This tendency tone "tends" to resolve up by a minor 2nd to Do (the leading tone leads to tonic). It is expected that the leading tone and the note it resolves to will be in the same musical voice.
Leading tones are found in dominant-functioning chords including V,V6, V7 (and its inversions) and vii°6 and vii°7 (and its inversions). In primary, dominant-functioning chords the leading tone is either the 3rd of the chord if it's a dominant triad or dominant seventh chord, or it's the root of the chord if it's a diminished triad or fully diminished seventh chord. In either case, the note is the leading tone and the normal resolution is up by minor 2nd.
This tendency tone also exists in applied (or secondary) chords. In these cases, the situation is the same in the context of the note being tonicized. The secondary leading tone is either the 3rd of the chord—if it's a dominant triad or dominant seventh chord—or it's the root of the chord—if it's a diminished triad or fully diminished seventh chord. In terms of the leading tone being scale-degree seven in a major key, however, it is now in relation to the note being tonicized. If the the note being tonicized is D, the leading tone is C♯, if the note being tonicized is B♭, then the leading tone is A.
In terms of voice leading, you'll want to be specifically aware of the location of all tendency tones in order to model normal resolution in European classical music of the "common-practice period" (approximately 1600-1900). Each leading tone (or secondary leading tone) moves (or resolves) up by a minor 2nd in the same musical voice. There are times when the leading tone moves instead down by half step but remains the same letter name (augmented unison). Please see the section below on elided resolutions for more information.
These terms are used to describe a less common phenomenon where the leading tone doesn't resolve up by minor second and instead leaps down a third to become the fifth of the chord of resolution. This typically only happens in inner voices, and even when it does happen, some other voice will have the pitch-class that the leading tone would normally have resolved to. Its resolution in these situations can be considered an "implied" resolution (see section below on implied resolutions for more information). When completing voice-leading exercises in this style, however, you should only do this when needed, not as an equal option. At perfect authentic cadences, in particular, there should be no reason to do this.
Chordal sevenths (e.g., B♭ in a C7 chord) tend to resolve down by step (either a major 2nd or minor 2nd, depending on the context). It is expected that the chordal seventh and the note it resolves to will occur in the same musical voice.
Altered pitches tend to resolve by step in the direction they were altered. So, D♯ in the key of C is the altered version of D♮ so it has been altered from its diatonic state. Because the note has been raised from its diatonic state, its tendency will be to resolve up to the next note in the scale, which in this case would be E. Still in the key of C, D♭ is the lowered version of D♮ so it has been lowered from its diatonic state. Because the note has been lowered from its diatonic state, its tendency will be to resolve down to the next note in the scale, which in this case would be C. It is expected that the altered note and the note to which it resolves will occur in the same musical voice.
The tendency of an altered note is based on how the note has changed from its diatonic state, not whether or not there is a sharp, flat, or other accidental. So, in the key of B♭, E♮ is the raised form of the diatonic note E♭, so E♮ would have a tendency to resolve up to the next note in the scale (F). Similarly, in the key of B minor, a C♮ is the lowered version of the diatonic note, C♯, so it's tendency would be to resolve down by step to B.
The resolution of a leading tone is often elided when a dominant-functioning chord is followed, not by a triad, but by another dominant-functioning chord with the same root (e.g., expecting G7 to C but get G7 to C7 instead). In these situations, the leading tone will often move down by step but retain its letter name. So, the leading tone in a G7 chord is B (which would normally resolve up to C), but in an elided resolution to the chord C7, B would move down to B♭ instead of resolving up to C.
Note: Remember that this page is about the tendencies (i.e., what tends to happen most often) of a particular style of music. In particular, the elided resolution mentioned above is the norm in many jazz styles, so it would NOT be considered elided in that style, it would simply be the normal tendency in that context.
Implied means that the musical situation gives the impression that it is a version of some common paradigm, yet it is missing an expected component of the paradigm being compared. It's as if the passage reminds the analysts of a paradigm even though it's demonstrably different. In the case of tendency tones, the expected resolutions aren't literally happening as expected but the musical affect is basically the same as if they were. This usually just means that the resolution of a tendency tones happens in a different musical voice than the one that had the actual tendency tone.