A plasmid is a molecule of double stranded, circular DNA which must have a replication origin in order for it to replicate in the cell independently and a selection gene so that it is not lost from the body during the course of many cell multiplications . Plasmids are found in prokaryotic organisms (bacteria) and in eukaryotes (including yeasts).
The plasmids used in biomolecular research may be far more complex and in particular also contain cloning cassettes which are short sequences recognised by restriction enzymes able to cleave DNA at these sequences.
In nature, plasmids are pieces of DNA which can be transmitted from cell to cell by horizontal transfer. If it carries a gene of interest (for example resistance to an antibiotic), it confers a clear advantage to cells which have it.
Plasmids (in blue) are circular pieces of DNA, which are dissociated from the chromosome (in red). © Spaully, Wikimedia, CC by-sa 2.5