General:
* Snippet
* Scrap
* Piece
* Bit
* Portion
* Section
* Fragment
* Residue
* Remnant
More specific:
* Shard (for a broken piece of something, especially glass or pottery)
* Splinter (for a small sharp piece of wood)
* Chip (for a small piece broken off something)
* Crumb (for a small piece of food)
* Tatter (for a small piece of cloth)
* Stubble (for a small bit of something left behind, like hair or grass)
Literary:
* Vestige (for a trace or small remaining part of something that has disappeared or ceased to exist)
* Relic (for a surviving fragment of something from the past)
The best word will depend on the context of your sentence and the specific object you are describing.