Suspended chords, and what is left when the third goes
The third replaced by the fourth or the second, which leaves the chord neither major nor minor; it almost always resolves to the third straight after, and when it does not it becomes a colour of its own.
The third decides whether a chord is major or minor. Take it out and put another note in its place, and the chord stops being either: it hangs suspended, with no settled colour, waiting for the third to come back. That is what a chart calls "sus", and it is one of the most recognisable sounds of popular guitar.
Sus4 and sus2
Two notes usually take the third's place: the fourth, a semitone above the major third, and the second, a tone above the root.
has C, F and G. has C, D and G. Neither has an E or an E♭, so neither is major or minor; the fifth is still perfect, and the chord sits open, almost hollow.
Notice that the two use the same notes in different orders: C, F, G is the sus4 of C, and F, G, C is the sus2 of F. The chart decides which is which by the bass.
The resolution
The name comes from "suspension", and a suspension asks to resolve: the fourth falls to the third, and the chord becomes what it was going to be.
That fall of a semitone, F to E, is one of the oldest gestures in Western music, and the guitar makes it with one finger: hold the C chord, put the finger on the F, take it off. It is the same movement in the sus4 of any chord, and it holds for sus2 too, where the second rises to the third.
On the dominant
The commonest home of the sus4 is the dominant chord. G with a seventh and a suspended fourth has G, C, D and F: the fourth takes the B's place, and the chord loses its tritone. It is dominant without the dominant's edge, and the resolution happens in two stages: the fourth falls to the third, then the whole chord resolves to the tonic.
It is the cadence of half the ballads of the seventies and eighties. Charts write G7sus4, or just G7sus, or sometimes Gsus; all of them mean the fourth in the third's place.
When it does not resolve
The third does not always come back. A sus2 or a sus4 left as it is becomes a colour of its own, open and without gravity, and there are whole genres that prefer that sound to the "complete" chord. On the guitar, chords with open strings produce sus2 and sus4 almost by themselves, and a good part of what gets called "the folk sound" is exactly this.
What to take from here
A suspended chord is a triad with its third swapped for the fourth or the second, which is why it is neither major nor minor. It almost always resolves, the fourth falling or the second rising to the third, and that resolution is the gesture that defines it. On the dominant, the sus4 removes the tritone and softens the cadence; left unresolved, it becomes an open colour the guitar produces with ease.