It's Genus species Author. Simple really.
Every now and then somebody asks me what the "L." means behind, for instance, Illicium anisatum.
That "L." means that Linnaeus was the first person to give that particular plant that particular botanically valid combination of genus and species.
Generally, it's enough to just keep track of genus and species, but sometimes the author is incredibly important.
To illustrate, have a look at, well, Illicium anisatum, just to take an example completely at random.
There's the edible Chinese species, and there's the toxic Japanese species. They look pretty much the same, too, and make similar fruit.
And both have been called Illicium anisatum.
So that's one instance where you have to know the author in order to know which plant we're talking about, in seeds, bushes, or whatnots.
The toxic Japanese star anise is still called Illicium anisatum L.; Linnaeus gave the name, and was the first to do so. The plant has had other botanical names, among them Illicium japonicum Sieb. and Illicium religiosum Sieb. & Zucc.
Siebold, a German who simply loved Japan, was thrown out of the country as a persona non grata, subsequently returned as a "northern Dutch" scientist, and was thrown out again, but I forget the details. Ah yes, gotta love the internet. There's obviously more to the story than that, though.
Zucc. stands for Joseph Gerhard Zuccarini. I've typed the combination of "Sieb. & Zucc." dozens if not hundreds of times, and have always thought of the pair as Siebold and Zucchini ... oh well, I'll continue to amuse myself with the zucchini.
The edible Chinese star anise is now called Illicium verum, and Hooker jr. (Hook. f., short for Hooker filius), was the first to give that botanically valid (as in, it's an Illicium) name to that particular species. It used to be called Illicium anisatum Lour., but Juao de Loureiro thought of the name too late - Linnaeus had gotten there first, and as L. meant a different plant, Lour.'s is just a synonym, these days.
Oh, and the italics and uppercases? It's Genus species Author.
For a list of some (but not all) botanical authors, check this page.