For example, the set of integers 1 to 3 with a depth of 1 the combinations would be.
[1]
[2]
[3]
While the combinations for the same set for a depth of 2 would be
[1,1]
[1,2]
[1,3]
[2,1]
[2,2]
[2,3]
[3,1]
[3,2]
[3,3]
It's is not quite a permutation, as the number can be reused.
This question is very similar to the recent question Name of $\{(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)\}$ in terms of $\{1,2,3\}$, but you're asking for names in general, so I'm not quite sure it counts as a duplicate.
$(1,3)$ is an "ordered pair". $\{(1,1),(1,2),(1,3),(2,1),\ldots\}$ is "the set of all ordered pairs of numbers in the set $\{1,2,3\}$". This set can be written with the cartesian product as "$\{1,2,3\}\times\{1,2,3\}$" or the cartesian power as "$\{1,2,3\}^2$".
$(3,1,2)$ is an "ordered triple" or an "(ordered) tuple of length $3$" or "a $3$-tuple". The set of all such things could be denoted "$\{1,2,3\}^3$" or (more arguably) "$\{1,2,3\}\times\{1,2,3\}\times\{1,2,3\}$".
In general, what you're calling "combinations for the [set $S$] for a depth of $n$" would be $n$-tuples (even if $n$ is $1$ or $0$), and the set of them could be written $S^n$.
Your special case of the set being something finite $\{1,2,3\}$ comes up often enough that there are other words/notations that can help. Wikipedia reports that in certain contexts, these tuples might be called "arrangements with repetition" or "permutations of a multiset". Some texts on Combinatorics use a special notation like $[m]$ to denote the set $\{1,2,\ldots,m\}$, so that the set of $n$-tuples would simply by $[m]^n$ and the number of them would be $m^n$.
