Fundamental representation and adjoint representation Why fundamental representation and adjoint representations of Lie algebra are very important in physics?
What about the other representations of a given Lie algebra? Are they equally important or not?
 A: Often$^1$ if we consider a Lie group $G$ that contain the symmetric group $S_N$ as a finite subgroup, such as e.g., $$GL(N,\mathbb{F}), \qquad U(N), \qquad \text{or}\qquad \{M\in U(N) | \det M =\pm 1\},$$ where $N\in \mathbb{N}$, with corresponding Lie algebra
$$gl(N,\mathbb{F}), \qquad u(N), \qquad \text{or}\qquad su(N),$$
respectively, then the finite-dimensional irreducible representations (irreps) are classified by Young diagrams. The trivial representation is no boxes; the $N$-dimensional defining representation$^2$ is a single box; and all other  finite-dimensional irreps can be understood as an appropriate tensorial generalization of the defining representation. So one could argue that the defining representation is the most important in that sense.
References:

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*P. Cvitanovic, Group Theory: Birdtracks, Lie's, and Exceptional Groups.

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$^1$ If we try this for $O(N)$, we would miss the (projective) spinor representations.
$^2$ For $SU(N)$, the defining representation is the fundamental representation. The adjoint representation of $SU(N)$ has a Young diagram of $N$ boxes
$$\begin{array}{rl} [~~]&[~~]\cr [~~]\cr \vdots \cr [~~] \end{array}$$
A: The two representations you mentioned are some that occur often, but they surely are not the only relevant ones. The fundamental representation is the defining representation. It is related to the group elements as matrices themselves, so, in this sense, it is quite important. It is also the representation that transforms "as a vector" (as opposed to a tensor).
The adjoint representation is the representation under which gauge bosons, for example, transform in gauge theory, which is also particularly interesting in Physics.
Examples of other representations occurring in particle physics
The history of QCD probably has some of the nicest examples of how representation theory can be used in Physics. I'm not an expert, so take this section with a grain of salt, especially when it comes to the history details.
In QCD, the quarks transform in the $3$ representation of $SU(3)$ (i.e., the fundamental), while antiquarks transform as $\bar{3}$ (the conjugate representation). Hence, mesons, which are composed of a quark and an antiquark, will transform according to the representations that occur in
$$3 \otimes \bar{3} = 8 \oplus 1.$$
Gell-Mann noticed in the sixties that the pseudoscalar mesons did organize in this way. There was an octet of mesons with sort of similar properties (the three pions, the four kaons, and the eta meson), and a singlet (the eta prime meson).
Furthermore, baryons, composed of three quarks, should transform according to
$$3 \otimes 3 \otimes 3 = 10 \oplus 8 \oplus 8 \oplus 1.$$
The neutron, proton, the three Sigma baryons, the two Xi baryons, and the Lambda baryon were known to form a spin-$\frac{1}{2}$ octet. No decuplet was known, but there were nine spin-$\frac{3}{2}$ particles with similar properties: the four Delta baryons, the three Sigma-* baryons, and  the two Xi-* baryons. Gell-Mann noticed these nine particles should be transforming in the $10$ representation, and hence there needed to be another one. About two years later, the Omega baryon was found, completing the decuplet.
