F0: The parents selected to start a breeding program. Often referred to as P1 and P2, but this is incorrect.
F1: the first cross between two unrelated parents. The F stands for filial, and refers to the fact that all F1 progeny of the same cross are full brothers and sisters to one another.
S1: The first selfed generation. Selfing an S1 produces an S2, etc. Anecdotal evidence from Sam_Skunkman indicates that continued selfing to the S3 and S4 produces plants so weak that they must be handled very carefully, “like kittens” in his words.
R1's (aka Reversed F1's): When feminized pollen is used to pollinate a different female than the pollen donor. R1's will tend to act like a tradional male x female cross, only all female, while S1's appear to have some different properties that are not yet fully understood. Early reports indicate that S1's are more consistent than R1's on average, but there are many exceptions, and more research is needed.
BC1: The first backcross generation, ie when an F1 or R1 progeny is crossed back to an F0 parent. Backcrossing can increase the influence of either parent, but continued backcrossing is too much inbreeding, according to both DJ Short and Rezdog, and should be used rarely if at all. One or two backcrosses followed by full-sib mating has beena successful strategy for many breeders, including the creator of Northern Lights.
So breeding goes like this:
F0(mother)+F0(father)=F1
F1+F1(same plant self pollinated)=S1
S1+S1(same plant self pollinated)=S2=weaker plant
F1+F1(different plants)=R1
S1+F0=BC1
These terms can be combined for shortand pedigrees. A second backross, followed by three generations of sib-mating, may be represented as a BC2-F3 generation.
Clear as mud?