"irritable" is usually used to mean "gets irritated" or "often gets irritated."
I personally suggest it's a "poor" word - don't use it. When you use it, you pretty much have to add an adjective like "very"; on the whole it's just much clearer to say "she gets irritated easily." You can (as you should) also be much more specific that way: "cats irritate her"; "everything at her job irritates her"; "almost everything irritates her".
"fractious" is little-used. When used, it is used to mean "starts arguments".
(Note that it comes from the word "fractions" - a person who tends to divide a group in to fractions through argument, a "splitter" if you will.)
Generally do not use it. It's almost certainly the case that you mean "argumentative." Only use "fractious" if (incredibly) you very specifically are trying to say "tends to cause divisions in to factions amount social groups. Of course, you'll never, ever need to say that in your life, so don't use the word.
"irascible" is only used when you want to sound like you know fancy words, so don't use it. If anything, it means "yells a lot," or, the "grumpy old man" type. Don't use it. Say what you mean (such as "He yells a lot" or "He always complains about dinner" or whatever the actual case may be.)
"atrabilious" is barely a word. Don't use it, forget about it.
"irritable" - gets irritated
"fractious" - a "splitter"
"irascible" - grumpy
"atrabilious" - non-word