Symbols have a variety of names, some functional, some historical, some humorous. For example, # might be called “number sign”, “octothorpe”, “fence,” etc. You can find an extensive list here.
Below is my abbreviated and idiosyncratic list, terms I use instinctively that may require some explanation.
! bang
| pipe
# pound
` back tick
^ caret, exponent
_ underscore, subscript
\ back slash, escape
/ (forward) slash
[ open (square) bracket
] close (square) bracket
{ open (curly) brace
} close (curly) brace