How does MySQL CASE work?

CASE is more like a switch statement. It has two syntaxes you can use. The first lets you use any compare statements you want:

CASE 
    WHEN user_role = 'Manager' then 4
    WHEN user_name = 'Tom' then 27
    WHEN columnA <> columnB then 99
    ELSE -1 --unknown
END

The second style is for when you are only examining one value, and is a little more succinct:

CASE user_role
    WHEN 'Manager' then 4
    WHEN 'Part Time' then 7
    ELSE -1 --unknown
END

CASE in MySQL is both a statement and an expression, where each usage is slightly different.

As a statement, CASE works much like a switch statement and is useful in stored procedures, as shown in this example from the documentation (linked above):

DELIMITER |

CREATE PROCEDURE p()
  BEGIN
    DECLARE v INT DEFAULT 1;

    CASE v
      WHEN 2 THEN SELECT v;
      WHEN 3 THEN SELECT 0;
      ELSE
        BEGIN -- Do other stuff
        END;
    END CASE;
  END;
  |

However, as an expression it can be used in clauses:

SELECT *
  FROM employees
  ORDER BY
    CASE title
      WHEN "President" THEN 1
      WHEN "Manager" THEN 2
      ELSE 3
    END, surname

Additionally, both as a statement and as an expression, the first argument can be omitted and each WHEN must take a condition.

SELECT *
  FROM employees
  ORDER BY
    CASE 
      WHEN title = "President" THEN 1
      WHEN title = "Manager" THEN 2
      ELSE 3
    END, surname

I provided this answer because the other answer fails to mention that CASE can function both as a statement and as an expression. The major difference between them is that the statement form ends with END CASE and the expression form ends with just END.

I wanted a simple example of the use of case that I could play with, this doesn't even need a table. This returns odd or even depending whether seconds is odd or even

SELECT CASE MOD(SECOND(NOW()),2) WHEN 0 THEN 'odd' WHEN 1 THEN 'even' END;