How do I introspect the class of a variable in APEX?

From p. 122 of the Apex Developer Guide:

If you need to verify at runtime whether an object is actually an instance of a particular class, use the instanceof keyword...

But, you cannot use the instanceof keyword on an instance of the the class of its subclasses or else you'll receive a compile error. For example:

Integer i = 0;
System.debug(i instanceof Integer);

>> COMPILE ERROR: Operation instanceof is always true since an instance of Integer is always an instance of Integer.

You need to use the instanceof keyword on superclasses only. For example:

System.debug((Object)i instanceof Integer);

>> true

If you ever need information on type of the Class itself, check the System.Type methods (pg 396 of current Apex developer's guide. Here's are some examples:

Type integerType;
integerType = Type.forName('Integer');
integerType = Integer.class;

Someone asked me this earlier today. Here is a more dynamic solution.

MyClass mine     = new MyClass();
String myStr     = String.valueOf(mine);
String className = myStr.split(':')[0];
System.assertEquals('MyClass', className);

A generic code of Salesforce Apex to explain :

public class Animal {}
public class Bird extends Animal {}

Animal a = new Bird();                      // Success
System.debug(a);                            // Bird:[]
System.debug((Bird)a);                      // Bird:[]
System.debug((Animal)a);                    // Bird:[]
System.debug(a instanceof Bird);            // true
System.debug((Animal)a instanceof Bird);    // true
System.debug((Bird)a instanceof Bird);      // Operation instanceof is always true since an instance of Bird is always an instance of Bird
System.debug((Bird)a instanceof Animal);    // Operation instanceof is always true since an instance of Bird is always an instance of Animal