identity: a variable holds the same instance as another variable.
equality: two distinct objects can be used interchangeably. they often have the same id.

Identity

For example:

Integer a = new Integer(1);
Integer b = a;

a is identical to b.

In Java, identity is tested with ==. For example, if( a == b ).

Equality

Integer c =  new Integer(1);
Integer d = new Integer(1);

c is equal but not identical to d.

Of course, two identical variables are always equal.

In Java, equality is defined by the equals method. Keep in mind, if you implement equals you must also implement hashCode.