A quick overview...
Music is made up of...
... and each layer can be assigned a role:
Melodic
Harmonic
Rhythmic
Sound effect
Texture can refer to the density (thick or thin) or the relationship (polyphonic, homophonic, monophonic) between the layers.
Here are some audio + visual examples of some of the most common textural relationships in music:
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Multiple melodies played at once
Melody with chordal accompaniment
One melody with no accompaniment
Multiple "voices" playing the same melody