Differences in boolean operators: & vs && and | vs ||

Those are the bitwise AND and bitwise OR operators.

int a = 6; // 110
int b = 4; // 100

// Bitwise AND    

int c = a & b;
//   110
// & 100
// -----
//   100

// Bitwise OR

int d = a | b;
//   110
// | 100
// -----
//   110

System.out.println(c); // 4
System.out.println(d); // 6

Thanks to Carlos for pointing out the appropriate section in the Java Language Spec (15.22.1, 15.22.2) regarding the different behaviors of the operator based on its inputs.

Indeed when both inputs are boolean, the operators are considered the Boolean Logical Operators and behave similar to the Conditional-And (&&) and Conditional-Or (||) operators except for the fact that they don't short-circuit so while the following is safe:

if((a != null) && (a.something == 3)){
}

This is not:

if((a != null) & (a.something == 3)){
}

"Short-circuiting" means the operator does not necessarily examine all conditions. In the above examples, && will examine the second condition only when a is not null (otherwise the whole statement will return false, and it would be moot to examine following conditions anyway), so the statement of a.something will not raise an exception, or is considered "safe."

The & operator always examines every condition in the clause, so in the examples above, a.something may be evaluated when a is in fact a null value, raising an exception.

I think you're talking about the logical meaning of both operators, here you have a table-resume:

boolean a, b;

Operation     Meaning                       Note
---------     -------                       ----
   a && b     logical AND                    short-circuiting
   a || b     logical OR                     short-circuiting
   a &  b     boolean logical AND            not short-circuiting
   a |  b     boolean logical OR             not short-circuiting
   a ^  b     boolean logical exclusive OR
  !a          logical NOT

short-circuiting        (x != 0) && (1/x > 1)   SAFE
not short-circuiting    (x != 0) &  (1/x > 1)   NOT SAFE

Short-circuit evaluation, minimal evaluation, or McCarthy evaluation (after John McCarthy) is the semantics of some Boolean operators in some programming languages in which the second argument is executed or evaluated only if the first argument does not suffice to determine the value of the expression: when the first argument of the AND function evaluates to false, the overall value must be false; and when the first argument of the OR function evaluates to true, the overall value must be true.

Not Safe means the operator always examines every condition in the clause, so in the examples above, 1/x may be evaluated when the x is, in fact, a 0 value, raising an exception.