In what cases would you apply the order of operations' principle to a mathematical problem?

It would be applied when there is a mixture of operations which have different positions in the order. For example:

2 x 4 x 6 - 6 + 3 x 7

Multiplication comes before addition so, in this case, the 3 x 7 would be done before the addition. Putting in brackets shows the order this is calculated:

(((2 x 4) x 6) - 6) + (3 x 7)

There are various acronyms to help remember the order of operations where each letter is one of the operations and they are applied as the letters of the acronym are read from left-to-right:

PEMDAS: Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction

BIDMAS: Brackets, Indices, Divide, Multiply, Add, Subtract

BODMAS: Brackets, Order/Of, Multiply, Divide, Add, Subtract

Note that Divide and Multiply, Add and Subtract have the same order (in pairs), so when applying the acronyms above if Multiply/Divide occur together they are calculated left-to-right in the order they are met, similarly for Add/Subtract. To avoid this there is the acronym:

GEMorDSorA: Grouping, Exponents, Multiplication or Division, Subtracting or Adding

However, Division is the same as the Multiplication by the reciprocal, and Subtraction is the same as the Addition of the negative, so as long as Division is always done before Multiplication, and Subtraction always done before Addition, the correct answer will result. An acronym which provides this is:

BODTSAR: [do in order] Brackets, Order/Of, Divide, Times, Subtract, Add [to finally get the] Result.