@Byron's overview of major types of bacteria, shared above, is excellent. I believe that at some point he (perhaps it was someone else?) shared some specific information on the bacteria contained in various "instant cycle" cultures. I for one would love to see that again, if anyone can find it.
I earlier tried to track down the article (by Dr. Hovanec) that provides his test results/observations on the "cycling" products, but couldn't seem to, so I can only go from memory here. I will have another search later today. But he stated that having tried I think five or six of these products, none contain the correct nitrifying bacteria, but some of them did tend to speed up the cycling process by a few days. He reasoned, if memory is serving me correctly, that it was the introduction of "bacteria" regardless of species/purpose that helped get the "cycle" started.
Only his product, Dr. Tim's One and Only, does instantly "cycle" a tank; he says this, and so far to my knowledge no scientific evidence has come forward to disprove this, and I am sure if it were not true someone would have been overjoyed to point it out, so one must conclude he is correct. His original formula (which is different bacteria from this product) was sold to Tetra and is available as their SafeStart. He does not, to the best of my knowledge, purport that this is an instant cycling additive, but it has been proven to help establish the cycling.
There is also the issue of archaea. This is something that I did not mention in my article (written a decade ago now), and it should be added. Evidence indicates that the nitrifying bacteria initially colonize a new tank, but very shortly thereafter these disappear and are actually replaced by archaea, a different group of single-cell organisms, certain species of which (at least in freshwater systems) carry out the oxidation of ammonia to nitrite, and nitrite to nitrate.