This is one of those questions that doesn’t have that satisfying an answer. From a purely numerical point of view, it’s just some irrational number, which means that you can’t write it as a fraction. Plug it into a calculator and you get 1.77245385091, which is a decent approximation (see chris above for a better approximation).
Geometrically, pi is the area of a circle with radius 1, so asking for the square root of pi is the same as asking for the side length of the square whose area is the same as the circle’s area. I don’t know any pressing geometric reason to do this. However, it was a famous (unsolvable) problem for many centuries to “square the circle” with a ruler and compass, which basically boils down to constructing a line segment with length sqrt(pi) exactly (or to construct the square I described above). It’s a corollary of Galois theory that this can’t be done, since pi is transcendental. Other than that, I’ve never seen a need to calculate the number specifically.
So, chris is right. There’s not too much special significance to the number that wasn’t already there in a clearer form in pi itself. Was there any reason you were curious about it?