Step 1: Read the given condition.
We are told $A$ is a $3\times 3$ matrix with $AA^{T}=I_3$, where $A^{T}$ is the transpose and $I_3$ is the identity matrix. We must name this kind of matrix.
Step 2: Recall the inverse idea.
The inverse of $A$ is the matrix that gives the identity when multiplied with $A$. So if $A\,B=I$, then $B$ is the inverse of $A$.
Step 3: Match the pattern.
Compare $AA^{T}=I$ with $A\,B=I$. Here the role of $B$ is played by $A^{T}$. So the transpose is acting as the inverse.
Step 4: Write the key equality.
This means $A^{T}=A^{-1}$. A matrix whose transpose equals its inverse has a special name.
Step 5: Name it.
A square matrix with $A^{T}=A^{-1}$ is called an orthogonal matrix. Its rows are unit vectors that are mutually perpendicular, and the same is true for its columns.
Step 6: Rule out the others.
It is not singular, because a singular matrix has no inverse, but here an inverse clearly exists. It is not skew-symmetric or nilpotent, since those obey different rules.
Step 7: Conclude.
So $A$ must be an orthogonal matrix.
\[ \boxed{\text{Orthogonal matrix}} \]