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There are two main types of vectors, cloning and expression vectors. Cloning vectors are ideal for the generation of many copies of your gene. If the goal is to express the gene of interest, an expression vector is needed.
For cloning vectors, the key aspects are the copy number (depending on the ori), selectable marker, and the cloning sites. Generally, a high copy number is preferred. Note that when the gene is toxic for the cell, or when the plasmid is unstable, a low copy number might be favorable. The selectable marker allows for the identification of a positive transformant. Most of the time these will be drug-resistance markers, but also auxotrophic markers are used. It is important to check whether the vector contains a cloning site which is suitable for the insert. To date, most vectors contain a multiple cloning site, which makes it likely that the vector is compatible with the selected restriction enzyme.
Expression vectors contain some additional sequences related to the expression, e.g., promotor, ribosome binding site, terminator, tag or fusion protein. Some of these sequences are specific to the host organism. Therefore, the expression vector needs to be compatible with the chosen host organism (e.g. mammalian, insect, E. coli).