General:
* Patch
* Scrap
* Snippet
* Fragment
* Piece (this is simple but might not be specific enough)
Specific to fabric type:
* Ribbon (thin, long strip)
* Linen (from flax)
* Cotton (from cotton plant)
* Silk (from silkworms)
* Velvet (soft, plush fabric)
Specific to purpose:
* Rag (for cleaning)
* Bandage (for medical purposes)
* Taffeta (for lining)
* Satin (for decoration)
* Felt (for crafts)
Informal:
* Bit
* Piece of material
* Little bit of cloth
The best word will depend on the specific situation.