What the F?

Welcome to my explanation of the terms F₁ and F₂. I'm writing this in preparation for a future post in which I want to reference these terms -- I don't want to have to explain them in that post, and I think most people know roughly what they mean, but just in case, I'm writing this so I can link back to it for the sake of the confused
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When you make a hybrid between two different varieties or species or whatevers the first generation is the F₁ generation. The babies of the F₁s are the F₂s, children of the F₂s are the F₃s, et cetera.
One talks about F₁s and F₂s a lot because they have special characteristics, which basically boil down to this: If I cross two different things, the F₁s will all be the same, and the F₂s will all be different.

Some real life pictures of this: For my research in grad school, I'm working with a hybrid of two species of petunia: Petunia axillaris, which is white, and the red flowered Petunia exserta.

 The F₁ generation of this hybrid all looks the same: Very, very pale pink flowers:

While a bunch of F₂ plants look like this: A random mix of all different shades of pink.

But WHY are the F₁s all the same, and the F₂s all different? Let me use another example: A couple years ago I made a hybrid between two tomatoes. One, Matt's Wild Cherry is (as you might guess...) a cherry tomato. The other, Black Krim, is not a cherry tomato. The cherry trait is controlled by a single gene: Have one or more copies of the dominate version (C) of this gene, and you make cherry tomatoes. Have two copies of the recessive version (c) and you make big fruits. So I made my two tomatoes have sex, and they had a bunch of little F₁ babies. Each F₁ got a copy of each gene from each of the parents. From Matt's Wild Cherry they got the dominate version C, and from Black Krim, the recessive version c. So all the F₁ plants had one C and one c, and since C is dominate, they were all cherry tomatoes. I then crossed the different F₁ plants with each other to produce the F₂ generation. Now each F₁ has one copy of C and one of c, so when they make babies, they randomly choose just one of the versions of the gene (sometimes C, sometimes c) to pass on to the next generation -- meaning among the F₂ plants, some will happen to get a C from both parents, some will get one C and one c, and some two copies of c – making the F₂ generation a mix of cherry and full sized tomatoes.
The exact same process works for all the other genes in the tomato: There are genes for the dark color of Black Krim, genes for the sweetness and flavor of their fruit, how they grow, when they flower, and so on. All the F₁ plants are the same with exactly one version of each gene from each parent, but the F₂ will be a wild mix of all the different possible combinations of the different genes from the two parents.

And why should you care? Well, seed companies use this all the time to their advantage. When they are developing a new variety of tomato, they need it to be uniform – all the seedlings need to look the same (if you buy a packet of Early Girl seeds, and some plants came out as cherries, and others yellow, you wouldn't be too pleased, now would you?) The easiest way to make a variety uniform is inbreeding: Cross closely related plants each generation and you'll eliminate genetic variation resulting in perfect uniformity. But just as it isn't smart to marry your sister, inbred plants have problems (well, usually – some plants, like squash, are actually fine with it). So what to do? Well, if you take two uniform, inbred lines, and hybridize them the F₁ generation will be not be inbred, because it is a hybrid, but will be uniform because it is a F_1. Problem solved. And even better, if a company sells F₁ hybrids, you have to buy new seeds every year because if you save your own seeds, they will produce the F₂ generation when all chaos breaks loose. So F₁ hybrid varieties are not only an easy way to make healthy, uniform varieties, they also are a good way to ensure repeat sales (Though to be honest, people saving seeds isn't much of a concern to seed companies – the bigger issue is other seed companies using their varieties to develop similar, competing varieties, something releasing F₁ hybrids makes that much harder to do.)

So that's it, really: The numbers after the F indicate what generation you are talking about, and the ones we generally care about are the F₁s, which are all the same, and F₂s, which are all different. I hope you now feel extremely wise.