The 'square root' sign is formally the 'radical symbol'.
'Brackets' are formally 'parentheses'
Is there a formal term for the 'fraction sign' or say the 'absolute value bars'?
EDIT: I can see some downvotes. It would be helpful if I could receive some feedback as to why this question is not wanted as to not ask similar questions in the future. A vote has been cast as 'Unclear' but I would think that my question is a relatively simple question. I'm not quite sure how it is 'unclear'.
Brackets really refers to a whole class of paired symbols, $()$ and $[]$ being only the most common. And as I said in the comments, the word by itself actually means different sorts in different dialects. (See Wikipedia's article.)
While the symbol $/$ is variously called the slash, forward slash, solidus, and so on, the symbol in a vertically typeset fraction seems to just be called the fraction bar (and no doubt used to be typeset using dashes and lines of various lengths: printers have always hated mathematicians and their weird symbols and propensity for requiring fairly complicated typesetting). The symbol $\div$ is called the obelus.
In general, Wikipedia has a large proportion of all the information you want to know about typographical symbols. (My favourite is the one about how to use the various dashes, or the Manual of Style article on the same, which is far more complicated than you'd expect. And you've almost certainly used the n-dash wrongly in the past.)