Mathematical:
+(plus), - (minus), / (divided by), and * (multiplied by)
Comparison:
> (greater than), < (less than), == (equal to), and != (not equal to)
Boolean:
&& (true if both operands are true), || (true if at least one operand is true), xor (true if ONLY one operand is true), and ! (true if a single operand is false)
Mathematical operators are exactly what they are called, they apply mathematical functions to the operands. Comparison is also pretty straight forward, they compare one operand to another operand. Boolean however may need a little more explaining.
Boolean is an extremely simple form of logic. In Boolean every statement is either True or False. Think of a light switch, it must either be turned on or off, there is no in between. Let me give you an example:
$a = true;
$b = true;
$c = false;
$a && $b;
This is asking for $a and $b to both be true, since they are both true, this expression is TRUE
$a || $b;
This is asking for $a or $b to be true. Again this is a TRUE expression
$a xor $b;
This is asking for $a or $b, but not both, to be true. Since they are both true, this expression is FALSE
! $a;
This is asking for $a to be false. Since $a is true, this expression is FALSE
! $c;
This is asking for $c to be false. Since that is the case, this expression is TRUE
