What is the use of the @ symbol in PHP?

It suppresses errors.

See Error Control Operators in the manual:

PHP supports one error control operator: the at sign (@). When prepended to an expression in PHP, any error messages that might be generated by that expression will be ignored.

If you have set a custom error handler function with set_error_handler() then it will still get called, but this custom error handler can (and should) call error_reporting() which will return 0 when the call that triggered the error was preceded by an @...

It suppresses error messages — see Error Control Operators in the PHP manual.

The @ symbol is the error control operator (aka the "silence" or "shut-up" operator). It makes PHP suppress any error messages (notice, warning, fatal, etc) generated by the associated expression. It works just like a unary operator, for example, it has a precedence and associativity. Below are some examples:

@echo 1 / 0;
// generates "Parse error: syntax error, unexpected T_ECHO" since 
// echo is not an expression

echo @(1 / 0);
// suppressed "Warning: Division by zero"

@$i / 0;
// suppressed "Notice: Undefined variable: i"
// displayed "Warning: Division by zero"

@($i / 0);
// suppressed "Notice: Undefined variable: i"
// suppressed "Warning: Division by zero"

$c = @$_POST["a"] + @$_POST["b"];
// suppressed "Notice: Undefined index: a"
// suppressed "Notice: Undefined index: b"

$c = @foobar();
echo "Script was not terminated";
// suppressed "Fatal error: Call to undefined function foobar()"
// however, PHP did not "ignore" the error and terminated the
// script because the error was "fatal"

What exactly happens if you use a custom error handler instead of the standard PHP error handler:

If you have set a custom error handler function with set_error_handler() then it will still get called, but this custom error handler can (and should) call error_reporting() which will return 0 when the call that triggered the error was preceded by an @.

This is illustrated in the following code example:

function bad_error_handler($errno, $errstr, $errfile, $errline, $errcontext) {
    echo "[bad_error_handler]: $errstr";
    return true;
}
set_error_handler("bad_error_handler");
echo @(1 / 0);
// prints "[bad_error_handler]: Division by zero"

The error handler did not check if @ symbol was in effect. The manual suggests the following:

function better_error_handler($errno, $errstr, $errfile, $errline, $errcontext) {
    if(error_reporting() !== 0) {
        echo "[better_error_handler]: $errstr";
    }
    // take appropriate action
    return true;
}