Electrons are fermions, and only one fermion can occupy a quantum state at a time - unless some other property is different (like spin, represented in the joke as the opposite sides of the table). This is why you can only have two electrons in an orbit around an atomic nucleus. Bosons operate under no such restrictions, which makes the joke funny, if you already know all of that. If not, well, analyzing humor is like dissecting a frog: few are interested, and the frog dies of it.
Fermions are particles of matter, while bosons (well, gauge bosons, anyway) are force carriers, like photons (which transmit the electroweak force) and gluons (which transmit the strong force). Here's a table:
It's almost like the periodic table, but without so much periodicity.
OK, so you see how photons and gluons have no mass, but the other ones all do? Well, the Standard Model of physics has a real hard time explaining how they get that way. So hard a time, in fact, that we kinda-sorta had to make shit up. I say "kinda-sorta" because, technically, the Standard Model predicts a solution, instead of merely pronouncing one (as religions are wont to do). This solution is called the Higgs mechanism, which would do the things we need it to do if in fact reality works this way, but it relies on a field called - can you guess? - the Higgs field.
Fine and dandy, yeah? Except there's a snag: see, in quantum field theory, fields are always associated with a particle (which is simply an oscillation in the field). So a photon is an oscillation in the electroweak field, and it transmits that electroweak force from place to place - it's the electroweak boson, if you will. So if there's a Higgs field responsible for the Higgs mechanism responsible for giving mass to elementary particles, then when that field oscillates, we ought to see a Higgs boson, right? Except that we've never seen one. And it turns out they'd be damned hard to see anyway, since even though they ought to be hundreds of times as massive as a proton, they decay in under a yoctosecond. And that's the smallest SI prefix we've taken the trouble to invent.
This is why CERN, with all of its crazy lasers, was built in the first place: to try and see if we could tease a Higgs boson into existence, which we'd only be able to tell anyway by carefully documenting the resulting explosion and seeing if it looks like what we think a decaying Higgs boson ought to look like.
'Bout like that.
And now it looks like they've done it. They're being really cagey about it, and I can't really blame them. I mean, everyone remembers that whole neutrino debacle, right? The one that looked like it might have been caused by forgetting to take relativity into account, but was actually caused by a loose cable? So yeah, fair play for making sure you've got all your ducks in a row and taking the thoughtful approach instead of the sensationalist one.
Fuck Jonathan Goldsmith, he is the most interesting man in the world!
So there we have it, or so it would seem. On a final note, I found out while brushing up on my physics how it came to be called the god particle: "Lederman said he gave it the nickname 'The God Particle' because the particle is 'so central to the state of physics today, so crucial to our understanding of the structure of matter, yet so elusive,' but jokingly added that a second reason was because 'the publisher wouldn't let us call it the Goddamn Particle, though that might be a more appropriate title, given its villainous nature and the expense it is causing'."
For further reading, see Alan Boyle's Cosmic Log for a really great infographic and some sweet videos.