Plasmid is an extra-chromosomal DNA in bacteria which is self replicating as it contains origin of replication. Hence they are able to multiply by themselves independent of host chromosomal DNA. It is one of the types of cloning vector used for cloning of genes. Cells with cloned genes are then preserved for future use.
An expression vectors also called as an expression constructs are usually a plasmids but their function are different from cloning vectors. They are generally used to introduce a specific gene into a target cell. Once the expression vector gets introduced inside the target cell, then it (vector) starts producing or expressing the protein product that is encoded by the gene it carries. Vector use transcription and translation machinery of target cell. An expression vectors are used to produce many recombinant protein.