Origins of replication
Every plasmid has one essential feature: the origin of replication. To make a circular, double stranded DNA replicate in E. coli, the origin of replication is the only thing you need. If that’s all you have, you’d call the thing a “cryptic” plasmid—which just means it replicates in the cell but has no selectable marker. However, there are many different origin types, and different ones are used for different things. Rather than referring specifically to the origin, most people will refer to “types of plasmid”. Usually, what they are talking about is just what origin "part" is present in the plasmid, but some "types of plasmid" are a combination of parts needed for replication. I'll go through the most common ones here.
A really important concept in thinking about plasmids is compatibility groups. In a nutshell, if you put two different plasmids into the same cell with the same origin of replication, they will compete for replication resources to their detriment. Sometimes this will result in one "winning out" over the other, other times they will stably replicate alongside one another. Even if they appear to be stable, you are likely to get fluctuations in the relative copy number of the two plasmids and can see strange affects on expression levels. So, putting two plasmids with the same origin into the same cell is something you want to avoid. The only trick with this is that you may *think* two plasmids are different origins, but they are really in the same compatibility group. So, be careful and make sure you know what group your plasmids are in--you can't just look at the name of the plasmid and assume they are different! Each of the origin types I describe below are distinct compatibility groups. In theory, you can put one of each into the same cell and get stable replication. This does not mean they will necessarily give the same copy number as if they were individually transformed, and some combinations are better than others.
The colE1 origin
Many folks who do DNA work never get beyond the colE1 origin. Plasmids with this origin are often referred to as pUC plasmids. They are also sometimes referred to as pMB1 plasmids. The pSB1A2 and pBca9145 plasmids are both colE1 plasmids. The extreme majority of commercial protein expression plasmids are also colE1 plasmids. If you ever happen upon using M13 phagemids or cosmids, these are also usually colE1 plasmids. The desirable characteristic of these plasmids are that they are "high copy". The copy number of a plasmid is simply how many molecules of plasmid are typically present within the E. coli cells. For colE1 plasmids, this is typically around 200 per cell, but it can easily change with growth conditions or other features present on the plasmid. Very similar to the colE1-derived plasmids are the pBR322-derived plasmids. pBR322 has another replication-related part, called ROP, which regulates the copy number to a more modest ~50 copies per cell (don't quote me on that number!). Some folks still refer to pBR322 as "high copy", others refer to it as "medium copy". If you delete the ROP gene in pBR322, you get a pUC-like high copy plasmid. Some common vectors are this instead of bonafide pUC-derived plasmids, but they behave pretty similarly. Common plasmids in this compatibility group are pUC18, pUC19, pProTet, pBR322, pSB1A2, pBca9145, pET vectors (Novagen), pBADmychisA (Invitrogen), pCR2.1 (Invitrogen TA kit), pGEX and pMAL.
The p15A origin
The p15A origin is the most commonly used medium-copy origin at something like 15-25 copies per cell. It is frequently used in combination with pBR322-derived plasmids as a nice and stable 2-plasmid system. Although it works great as a two-plasmid system, putting colE1 and p15A origins on the same plasmid makes for a pretty unhappy cell. When working with p15A plasmids, the most important thing to keep in mind is that you are going to get dilute minipreps. You can still often sequence and perform in vitro manipulations on p15A plasmids, but it can be tricky.
The pSC101 origins
The pSC101-derived origin can coexist in three-plasmid systems with a p15A and colE1 origin stably. They are about 10 copies per cell making them pretty tricky to work with. You will likely either need RCA or PCR amplification to sequence off them. You can, however, put pSC101 and colE1 origins into the same plasmid, and it will stably replicate as a high copy plasmid.
The R6K and oriV origins
Some origins require additional genes in order to be functional. These can be incredibly useful for making things like suicide vectors or genome integration cassettes. The two most common are R6K and oriV. R6K is the more popular of the two. It requires expression of a gene called pir encoding pi protein to be functional. Wild-type E. coli does not have this gene, so the R6K origin doesn’t do anything in most strains. A variety of strains, my personal favorite being the Ec100D strains from Epicentre, have either the pir+ or pir116 genes in their genome and can replicate R6K origins. When using a pir+ strain, R6K-containing plasmids replicate just like medium-copy p15A plasmids. In pir116 strains, they behave as high-copy pUC plasmids. As long as you aren’t turning it on with pir, you can put R6K into a plasmid along with any other origin. In a pir strain, however, some combinations work and some don’t—in general, the lower the copy number of the “other” origin, the more stable the plasmid will be. In terms of compatibility group, it is distinct from the others. In practice, putting a colE1 and R6K plasmids into a pir116 cell leads to some unhappy cells. The story is pretty much the same for an oriV origin. It is derived from RP4 plasmid and requires the trfA gene for replication.
The F plasmid and BACs
Many common strains of E. coli, including JM109, XL1 blue, and TG1, already have an F plasmid in them. The original F plasmid is conjugative meaning it gets transferred by mating from cell to cell. Most of the common lab strains have mutations in one of the genes required for conjugation and are not competent for conjugation. Some common laboratory bits and pieces are derived from F plasmid including the ccdB gene (a counterselection marker often used in Biobrick and Gateway cloning) and bacterial artificial chromosomes (BACS). They also play an important role in filamentous phage (like M13 phage) infections involved in things like phage display and kunkel mutagenesis. Just keep in mind that if you are going to do any of these things, you should think about whether your strains have F plasmid or not. For synthetic biology purposes, the thing you most want to know about is BACs. They can be wonderful tools for doing genetic devices. Since they require the same replication machinery as F plasmids, you can’t put a BAC into a cell with F plasmid. One will kick the other one out-- the incompatibility behavior is particularly strong for the F plasmid origin. The reason for this is probably that they are very tightly replicated as one genome-equivalent or so per cell. The actual origin of replication for BACs and F plasmids is called oriS, but there are several additional parts needed to make a functional plasmid. The repE gene encodes a protein that plays a similar role to what pi does for an R6K origin. The parA, parB, and parC help the plasmid stably partition to daughter plasmids during replication. So, altogether, you need about 5kb worth of DNA to encode a BAC. The great thing about BACs is that they stably replicate at as low a copy number as you can get without directly integrating your DNA into the E. coli genome. Also, you can stably maintain DNAs up to around 250kb on a BAC. You can have any of the other compatibility group plasmids in the same cell as a BAC (except of course another F plasmid). If you put a higher-copy origin into a BAC, you’ll get the higher copy number associated with the second origin. There are also many uses for copy-inducible variants of BACs. By putting R6K or oriV origins in the BAC, you can make systems that can be ramped down to single copy or ramped up to high copy by controlling the pir or trfA expression levels.
PACs are kindov like BACs, but they aren’t very common. They have replicons derived from P1 phage. mob and rep plasmids. There are a number of plasmids containing origins referred to as mob and rep. In the synthetic biology world, you’ll most likely run into this with plasmids derived from pPROBE-GFP[LVA]. The rep portion of these plasmids is the replication origin, the mob portion is a conjugative origin. If you see that a plasmid has mob and rep, do not assume it is the same mob and rep as every other mob/rep plasmid. There are several distinct plasmids in use that call features mob and rep. As for compatibility with other origins, there can be interferences with some of these, so proceed carefully! Natural isolates. There are all sorts of origins of replication out there. Some exist naturally as plasmids, others are phage-derived. I’ve just listed a few of the really common ones here. Just keep in mind that there are many more out there!