The absolute value of a number is the (positive) distance of the number from the origin, 0. In other words, if the number x is positive, the distance from the origin |x| is simply x - 0 = x. If x is a negative number, then the distance from the origin is 0 - x = -x, a positive number. In order to reconcile this with a single formula, we rigorously define the absolute value of a real number x to be |x| = \sqrt{x^2} which will always be positive, as the principal square root function is defined to be. Sometimes you will want to use this definition, other times it is easier to use the first definition:
|x| = \left\{\begin{array}{lr}x & ,x \geq 0\\ -x & ,x < 0\end{array}
The two definitions are equivalent.
Since you're considering the limit as x approaches -1 of |x+1|, which considers the behavior of the expression near |0|, you can either break it into left- and right-handed limits with the second definition, noting that the limit exists if and only if both left- and right-handed limits exist and are equivalent, or you can do the full limit with the square root definition, squaring the expression and studying the behavior of the principal square root as x approaches -1.
