Quarks have a variety of names (or flavours):
Why do they have such odd names?
[Physics] Why are the quarks so named
quantum-field-theoryquarksstandard-modelterminology
Related Solutions
It is important to remember that words like this get used in spit-balling sessions and then stick. You have to think of a couple of guy sitting by a black board, coffee in hand saying something like
OK, OK! SO that doesn't work. But what if we assume these things come in three flavors and ...
It's just a word made up on the spot.
That said, I think that "up" and "down" came from an analogy with spin. "Strange" interaction were so called because they violated rules of thumb for other known interactions...
After that things got out of hand.
Side note: "flavor" is also used to distinguish the generation of leptons (and will no doubt be applied to any sterile neutrinos that make an appearance).
Side note$^2$: For a long time there were competing names for the third generation flavors. I'm a little sad that "top" and "bottom" won out against "truth" and "beauty".
Yes, the 6 antiquarks are antiparticles of the 6 quarks – in other words, they're particles of "antimatter". The word "antimatter" sometimes represents just a relative label – antimatter of something (antimatter of antimatter is matter again), sometimes it means the antimatter of the particles we routinely see in the world around us.
Because the 6 antiquark flavors – anti-up, anti-down etc. – have the same properties as the quarks (up to the opposite signs), they're not counted as "independent types of elementary particles". Quite generally, we don't consider antiparticle species to be "independent species" because it's a completely general fact that every particle type has an antiparticle (although, in some cases such as the photon, Z-boson, or Higgs boson, they coincide with the original particle).
No one would ever say that there are "12 types of quarks" because of the antiquarks. We either consider antiquarks "not to be quarks" when we talk about "quarks" in a strict sense, or we do include antiquarks among quarks but the antiparticles are considered to be pretty much the same thing as the original quarks (despite the sign flip in all quantum numbers) which is why we still have just 6 quark flavors (the types are called flavors; each of them also has 3 colors and 2 spin polarizations).
Leptons are not composed of quarks. Leptons and quarks are two equally large but mutually disjoint sets of elementary particles – leptons plus quarks are known as "elementary fermions".
The four forces are mediated by the photons (electromagnetic), W-bosons and Z-bosons (weak nuclear force), gluons (strong force), and gravitons (the gravitational force). Physics is pretty much equally sure about all four or five of them. The only way in which gravitons differ is that gravity is such an extremely weak force that individual gravitons are pretty much undetectable. But they're detectable if they're coming in sufficiently strong beams or packages – gravitational waves – and the 1993 physics Nobel prize was given out for the evidence that gravitational waves existed exactly as predicted by Einstein's general theory of relativity.
The Higgs boson is a boson (i.e. not fermion) but it's the only boson in the list that doesn't mediate a fundamental force. It's still very important in the scheme of the Universe because it guarantees that W-bosons, Z-bosons, (charged) leptons, and quarks are massive – via the Higgs/BEH mechanism. The Higgs boson was discovered last July.
Quarks differ by their carrying a color - interacting via the strong force (one mediated by gluons and described by QCD). Leptons don't carry any color so they don't interact by the strong force – which is the reason why their name, "leptons", is related to words like "skinny" in Greek.
Best Answer
The up/down top/bottom should be self evident: in a matrix representation the vector is written that way
+1/2>
-1/2>
So in the isotopic spin space (the SU(2) of the SU(2)xSU(3)xU(1) of the Standard Model) according to the charge the one on top was called the up and the one on bottom, the down.
The strange got its name from the strange mesons, which when discovered were behaving strangely, with respect to pions, needing a new quantum number because they were generated in pairs ( Lamda K) and the quantum number became the "strange" one.
Charm was a whim as , they were charmed by its existence since it had been predicted to exist given the quark model expectations; from the quark entry in wikipedia:
Top and Bottom, again because of the position in the vector, and Beauty instead of Bottom out of whimsy again, keeping the B, and Truth keeping the T.
Who said that physicists are not having fun?
For completeness, the name "quark" has the origin in Finnegan's Wake of James Joyce:
Of course our Germanic language friends say that Quark is a type of cheese!
Actually the quark article in wikipedia has an etymology section for the quarks, to be read for completeness.