Failure is the act of failing (not achieving a goal, or "fracasso") or not doing something (failing to do something) that you were supposed to do. In the first sense, it's the opposite of "success". An unsuccessful person can also be described as a "failure".
Their efforts resulted in utter failure (= total lack of success).
Failure (= failing) to comply with the regulations will result in legal action.
The plane crashed due to an engine failure (the engines on the plane stopped working).
The distinction between "flaw" and "fault" is a slightly trickier one, as both words tend to assume largely overlapping meanings in a number of circumstances. If we were to resort to translation, it could be argued that a "flaw" is more like a "falha", while a "fault" is more like a "defeito". In other cases, collocation should govern the choice. Examples:
Uma falha de caráter = A character flaw
Uma falha de design = A design flaw
Uma falha na lógica de alguém = A flaw in someone's logic
Uma falha no plano = A flaw in the plan
Nossas virtudes e defeitos = Our virtues and faults
Amava-a apesar de seus defeitos = He loved her despite her faults
O produto apresenta diversos defeitos = The product has several faults
Um defeito mecânico = A mechanical fault
A product with a manufacturing defect is typically said to be "faulty". If a perfectly working product doesn't serve its purpose properly or effectively, it could be said to have a "flawed design".
In the context of personalities, a "flaw" tends to be more easily forgivable than a "fault". A person's incessant desire to please others could be said to be a "flaw", in the sense that it's the result of a good, well-meaning nature. An unnaturally bad temper, on the other hand, would most likely qualify as a "fault", in that it represents a perceptibly more negative aspect of someone's character.
These are not rules set in stone. There are many cases where both words can be used interchangeably. Try and look at the words in context to naturally assimilate their usage patterns.