Unless the calculator is in approximate mode, or you force it to approximate (by pressing ♦+ENTER, or using approx()), it won't try to evaluate all square roots: it will take the square root of perfect squares, otherwise, it will just take out all the square factors (for instance, √(20) is simplified to 2√(5)).
For positive numbers, √() will return the positive square root; more generally, if the result is complex (and if the calculator is in complex number mode), the result of √() will be the one with non-negative real part.
:√(16) 4 :√(-12) 2*√(3)*i
If the square root of a list is taken, it will take the square root of every element of the list.
The √() of a matrix is not (in general) the same as taking the square root of every element of the matrix. A different definition is used to compute the result; see Matrices and Their Commands. It requires the matrix to be square and diagonalizable in order to apply.
230 - Dimension happens when taking the square root of a non-square matrix.
665 - Matrix not diagonalizable happens when diagonalization (used to take square roots of matrices) fails.
800 - Non-real result happens when taking the square root of a negative or complex number, in real number mode.