What are the parts of a plasmid and their functions?

Im assuming you are talking about a plasmid to be used in genetic engineering.
If not then the parts of a plasmid are the origin of replication, which is the site where the replication fork starts, this is where a loop is made in the plasmid and the DNA is replicated, and the genes.

In genetic engineering the parts of a plasmid that are important are;

The origin of replication - As stated before this allows the plasmid to be reproduced in a bacteria, usually E.coli.

Ampicilline resistance gene - This is used to select for bacteria that have taken up the engineered plasmid. For example say you engineer some DNA into a plasmid and then you want to put the recombinant plasmid into E.coli, some of the bacteria will not take up your plasmid. So you treat them with the antibiotic ampicilline and the ones that survive are your recombinants.

Multiple Cloning Site - This is quite important, it is essentially a piece of DNA that contains many different recognition sequences for restriction enzymes. For example you may want a plasmid to be able to be cut by as many different restriciton enzymes as possible to allow you to insert as many different fragments of DNA as possible. A multiple cloning site may contain cuttin sequences for several enzymes, e.g BamHI, EcoRI, Xbal, etc.

This site is often located within a gene that codes for a easily distinguishable protein, this is helpfull as you will be able to tell if the plasmid is recombinant ( containing foreign DNA ) if the protein is no longer made in the colony of bacteria.

For example a widely used gene is the Beta Galactosidase gene,  this turns a galactosidase into a blue dye, so when you add DNA to your plasmids the ones that take up the DNA will no longer have a functional copy of this gene, as the cloning site is in the middle of the gene, so they will appear white instead of blue in the presence of the galactosidase.