The single ampersand & is the logical AND operator. The double ampersand && is again a logical AND operator that employs short-circuiting behaviour. Short-circuiting just means the second operand (right hand side) is evaluated only when the result is not fully determined by the first operand (left hand side)
&& and || are short circuit operators operating on scalars. & and | operate on arrays, and use short-circuiting only in the context of if or while loop expressions.
As already mentioned by others, & is a logical AND operator and && is a short-circuit AND operator. They differ in how the operands are evaluated as well as whether or not they operate on arrays or scalars:
& (AND operator) and | (OR operator) can operate on arrays in an element-wise fashion.
&& and || are short-circuit versions for which the second operand is evaluated only when the result is not fully determined by the first operand. These can only operate on scalars, not arrays.
&& and || take scalar inputs and short-circuit always. | and & take array inputs and short-circuit only in if/while statements. For assignment, the latter do not short-circuit.
A good rule of thumb when constructing arguments for use in conditional statements (IF, WHILE, etc.) is to always use the &&/|| forms, unless there's a very good reason not to. There are two reasons...
As others have mentioned, the short-circuiting behavior of &&/|| is similar to most C-like languages. That similarity / familiarity is generally considered a point in its favor.
Using the && or || forms forces you to write the full code for deciding your intent for vector arguments. When a = [1 0 0 1] and b = [0 1 0 1], is a&b true or false? I can't remember the rules for MATLAB's &, can you? Most people can't. On the other hand, if you use && or ||, you're FORCED to write the code "in full" to resolve the condition.
Doing this, rather than relying on MATLAB's resolution of vectors in & and |, leads to code that's a little bit more verbose, but a LOT safer and easier to maintain.