How to understand the makeup of neutral pi and eta mesons? I know that mesons are bosons made up of quark-antiquark pairs. But when I see the list of mesons, I can see that the makeup of neutral pions and eta mesons are noted in a strange way.
$$\pi^0=(u\bar{u}-d\bar{d})/\sqrt{2}$$
$$\eta^0=(u\bar{u}+d\bar{d}-2s\bar{s})/\sqrt{6}$$
How am I supposed to understand their compositions?
Interpretation 1: a neutral pion should be understand as a quantum superposition and is actually composed of 2 pairs, sometimes appearing as an up pair, some other times as a down pair.
Interpretation 2: a neutral pion can be an up pair or a down pair. Both compositions lead to mesons with the exact same characteristics and behaviours.
What is the meaning of those square roots? If it's too complicated to be explained within a few lines, can anyone recommend me a website or a book?
 A: I think your interpretation 1 is correct. Here is a way to understand the linear combinations:
A meson made initially from just $u$ and $\bar{u}$ will not stay that way for long, because the quarks can annihilate and then reappear as $d\bar{d}$ or $s\bar{s}$. 
However, certain superpositions of $u\bar{u}$, $d\bar{d}$ and $s\bar{s}$ will remain constant over time. These are the linear combinations you listed, and
they are the $eigenvectors$ of the system's Hamiltonian. 
They can be derived by writing down the Hamiltonian in the ${u\bar{u}, d\bar{d}, s\bar{s}}$ basis:
$$H = \begin{bmatrix}2m + A & A & A\\A & 2m+A & A\\A & A & 2m+A\end{bmatrix},$$
where $m$ is the mass of a quark, and $A$ is the coupling between the basis states - it is the amplitude for a pair to annihilate and reappear as a different (or the same) pair.
(We are assuming here that all quarks have the same mass and the same annihilation amplitudes. This is known as SU(3) flavor symmetry.)
You can check that the eigenvectors are the two you listed (with eigenvalue $2m$), along with $(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}})$,
with eigenvalue $2m + 3A$, which corresponds to the $\eta^{\prime}$ meson.
Note: As alluded to in the other answers, this whole discussion is in the context of the simple static quark model, which is a big simplification of the real physics.
A: One has to keep in mind that all these constituents of the hadrons are elementary particles, i.e.quantum entities. All mathematical expressions follow the rules of quantum mechanics.
Quantum entitities are expressed with normalized wave functions .  So one should read the expression as the wavefunction of a pi0, and the wavefunction of the eta. Wavefunctions 

will give the probability of finding an up quark or a down quark when scattering off a pion ,and a strange quark when scattering off an eta. The square roots come so that the probability is normalized to one.
Hadrons are more complicated than the valence quarks that characterize their symmetries. The proton does not have just the valence quarks but a sea of quarks and gluons due to the strong interaction between quarks.
 
It is not easy to scatter off a pion :), but the proton has been extensively studied . These scatters are what come up in the parton distribution functions within hadrons .


Figure 1: Overview of the CTEQ6M proton parton distribution at Q = 2 GeV (Pumplin et al. 2002).

As you see it is much more complicated sinc there are not only the valence quarks but also the sea of quarks and gluons with which incoming leptonic probes can scatter. The valence quarks are important in the assignment of the symmetry groups,


The meson octet. Particles along the same horizontal line share the same strangeness, s, while those on the same left-leaning diagonals share the same charge, q (given as multiples of the elementary charge).

The eightfold way symmetries clinched the existence of quarks by the prediction and subsequent discovery of the omega-.
A lot enters when one is really studying hadrons.
