It is a napkin that has appears to be used, hence its name.
Role in episode
After deciding to live indoors forever, SpongeBob chooses Used Napkin as one of his new friends. At one point, SpongeBob scolded the napkin by saying, "You can do it without the name calling, thank you very much, Used Napkin!" Used Napkin is one of three "friends" SpongeBob made after deciding to live inside. The other two were Penny and Chip.