Hint: Your notes are apparently describing the symmetry of the corresponding Young diagram for each $SU(3)$ irrep. Each box corresponds to an index. Roughly speaking, indices in same row (column) are symmetric (antisymmetric), respectively.

E.g. a single box corresponds to the fundamental irrep ${\bf 3}$. Two boxes on top of each other is the antifundamental irrep $\bar{\bf 3}$ if we dualize with the help of the Levi-Civita epsilon tensor, and so forth.

Since this question looks like homework we will be somewhat brief. OP's notes are apparently describing the symmetry of the corresponding Young diagram for each $SU(3)$ irrep. Each box corresponds to an index. Roughly speaking, indices in same row (column) are symmetric (antisymmetric), respectively.

Examples:

1. A single box $[~~]$ corresponds to the fundamental irrep ${\bf 3}$.

2. Two boxes on top of each other $\begin{array}{c} [~~]\cr [~~] \end{array}$ is the anti-fundamental irrep $\bar{\bf 3}$ if we dualize with the help of the Levi-Civita symbol $\epsilon^{ijk}$. Here we adapt the sign convention $\epsilon^{123}=1=\epsilon_{123}$.

3. The tensor product ${\bf 3}\otimes{\bf 3}\cong\bar{\bf 3}\oplus{\bf 6}_S$ corresponds to $$ [~~]\quad\otimes\quad[a]\quad\cong\quad\begin{array}{c} [~~]\cr [a] \end{array}\quad\oplus\quad\begin{array}{rl} [~~]&[a] \end{array}$$
   or $T^{ij}=\epsilon^{ijk}A_k+S^{ij}$, where $A_k:=\frac{1}{2}T^{ij}\epsilon_{ijk}$.

4. The tensor product $\bar{\bf 3}\otimes{\bf 3}\cong{\bf 1}\oplus{\bf 8}_M$ corresponds to $$\begin{array}{c} [~~]\cr [~~] \end{array}\quad\otimes\quad[a]\quad\cong\quad\begin{array}{c} [~~]\cr [~~]\cr [a] \end{array}\quad\oplus\quad\begin{array}{rl} [~~]&[a]\cr [~~] \end{array}$$ or $T^i{}_j=S\delta^i_j+M^i{}_j$, where $S:=\frac{1}{3}T^i{}_i$, and ${\rm Tr}M=0$.

5. The tensor product ${\bf 6}_S\otimes{\bf 3}\cong{\bf 8}_M\oplus{\bf 10}_S$ corresponds to $$\begin{array}{rl} [~~]& [~~] \end{array}\quad\otimes\quad[a]\quad\cong\quad\begin{array}{rl} [~~]&[~~]\cr [a] \end{array}\quad\oplus\quad\begin{array}{rcl} [~~]& [~~] & [a] \end{array}$$ or $T^{ij,k}=\left\{M^{i}{}_{\ell}\epsilon^{\ell jk}+(i\leftrightarrow j)\right\} +S^{ijk}$, where $M^i{}_{\ell}:=\frac{1}{3}T^{ij,k}\epsilon_{jk\ell}$, and ${\rm Tr}M=0$.

References:

1. H. Georgi, Lie Algebras in Particle Physics, 1999, Section 13.2.

2. J.J. Sakurai, Modern Quantum Mechanics, 1994, Section 6.5.