Are antimatter particles made of ---? Consider some particle P, and the antiparticle aP.
Is it the case that:


*

*aP is made of the "usual" elementary particles, and indeed the same elementary particles as P, but arranged (or something) in some different way?


or,


*aP is made of the "usual" elementary particles, but it is made of some different combination of elementary particles than P?


or,


*aP is in fact made of not the usual elementary particles, but, is made of anti-elementary particles?

 A: Sometimes, they are the same particles and are not "arranged in some different way", for example photons are their own antiparticles, as are the hypothetical Majorana fermions.
However the most common way that this happens is that every matter-particle, for example the electron, has an antiparticle (in this case, the positron). If you wanted to talk about the antimatter hydrogen atom, then, it would consist of a positron orbiting an antiproton:
The antiproton is made up of two anti-up quarks and one anti-down quark, just as the proton is made of two ups and a down.

(As indeed can easily be seen on Wikipedia)
Quark binding is actually interesting due to the existence of mesons. It turns out that color charge can be balanced in two different ways that we call "baryons" and mesons (which almost instantly decay). When a particle has three quarks, we call that a baryon: and when there's one quark and an antiquark, we call that a meson.
For example, the pions are mesons made from the same quark species as protons and neutrons: there is a $\pi^+$ made of an up and an anti-down, and a a $\pi^-$ made of a down and an anti-up, and they are antiparticles of each other. There are also two $\pi^0$ mesons, one with an up and an anti-up, and the other with a down and an anti-down, which are each their own antiparticles and, as you can imagine from containing a particle and its antiparticle together, decay almost immediately (they have something like a billionth of the lifespan of the $\pi^{\pm},$ which themselves only last for tens of nanoseconds).
A: To start with this is the table of elementary particles:

A corresponding table exists for antiparticles for the blue and the green colored ones, which are fermions. From the rest which are bosons the neutral ones are the antiparticles of themselves, and the antiparticle of  W+ is W- and vice verso.
By definition the antiparticle has the opposite in numbers quantum numbers of a particle. 
These are the building blocks of matter.
The resonances, the baryons, proton neutron, and the nuclei that are built up by baryons correspondingly have their antiparticles . 

(1) aP is made of the "usual" elementary particles, and indeed the same elementary particles as P, but arranged (or something) in some different way,

the antiparticles are made up by the elementary particles in  the antiparticle table.

(2) aP is made of the "usual" elementary particles, but it is made of some different set of elementary particles than P,

same answer as 1),

(3) aP is in fact made of not the usual elementary particles, but, is made of anti-elementary particles. 

Yes, antiparticles are as valid as particles, except we live in a universe where particles dominate, leading to the baryon asymmetry, the CP violation 

In particle physics, CP violation (CP standing for charge parity) is a violation of the postulated CP-symmetry (or charge conjugation parity symmetry): the combination of C-symmetry (charge conjugation symmetry) and P-symmetry (parity symmetry). CP-symmetry states that the laws of physics should be the same if a particle is interchanged with its antiparticle (C symmetry), and when its spatial coordinates are inverted ("mirror" or P symmetry).

It is still an open research problem to explain why our universe is made mainly out of particles, without resorting to the anthropic principle
