A false relation is a harmonic thing, and it's when there are two consecutive (or very nearby) notes in different voice parts, and those notes have the same letter name but different accidentals. For example, a B flat in the bass part followed immediately by a B natural in the tenor part. (it can also refer to different notes in the same chord, but consecutive chords are more common.)
The Wikipedia article gives an example from Byrd's Ave Verum Corpus. A more familiar one, though, might be the Mozart version. In the third bar of the clip below, you can see the false relation in the bass and tenor parts exactly as described in the paragraph above.
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