Based on playing Fallout 3 and none of the previous games, I'd say you're fine to jump right in. It's more a sequel in terms of setting than character. You'll almost certainly miss out on some in-jokes or references, but I can almost guarantee you won't need to have played previous games in order to understand this one.
That said, if you do want to better understand the story, there are series of videos on youtube you can watch to really get a deep understanding. Basically, in some alternate-USA where fashion and technology seemed stuck in the 1950s (but was really well into the future), there was a nuclear war. Large portions of the country became an irradiated wasteland. One company, Vault-Tec, set up hundreds of bomb-proof bunkers around the country and filled them up with survivors whose families have lived for generations now in those bunkers. Vault-Tec secretly monitored those bunkers and set up weird experiments in some of them to see how people would react. Outside the bunkers, the radiation caused weird mutations, and mutant people and animals roam around the countryside, along with more normal bandits and marauders. As far as I know, each story usually follows one bunker-dweller who leaves the bunker to do something (e.g., in Fallout 3, you were looking for your dad) and takes place in and around one city.
That's pretty much it. I doubt you really need to know more than that to pick up the new one and enjoy it.
Edited to add: Here's a video summary of the main events of Fallout 3, if you're interested:
