They can mean several things, depending on where they are found.
Most commonly, they are found when you want to pass information ("Parameters") into something else ("Functions", "Classes", etc).
Sometimes, functions or classes don't actually need anything passed into them. But the computer is too stupid to know this, so it will need the ()'s anyway. If you forget the ()'s, the computer might misinterpret what you wanted, or just plain fail.
Think of it like this, if you give the computer an order, it needs you to tell it the details, and the details are always in brackets. If there aren't any details - it needs to know that, too!
read_a_book(15); // Makes it read starting on page 15
read_a_book(); // Also makes it read, but the computer will assume it starts on page 1
()'s can also be used for math.
1 + 2 * 3 = 5 // Bedmas will make this try multiplying first
(1 + 2) * 3 = 6 // Brackets will make this add first, since bedmas starts with brackets!
And they can also be used to clear up confusion if it doesn't know how to do what you want:
"I have ".2+5."apples"; // This won't work. it might think you are adding "I have 2" and "5 apples".
"I have ".(2+5)."apples" // Works fine.
Brackets can also be used for several other things, like typecasting. But 99% of your programming will be these scenarios.
In the mysql_error example:
()'s will let the computer know when a function is a function, and not something else. For example, you can create something called a "definition". A definition is just a word with a value - like a variable. Only definitions cannot be changed, and do not require the $.
so I could go...
define("mysql_error", "HI!");
echo(mysql_error()); // Might output "SQL ERROR: You have an error in your..."
echo(mysql_error); // This would output "HI!", because I made that definition.
The ()'s let the computer know that we wanted to call a function, and not anything else. Even though there were no parameters, when you give a computer an order - it expects to know if there are any details - or it will assume you meant something else.