This is a surprisingly difficult question! The short answer is that proof by contradiction is only valid when the propositions involved are "well-formed" in a certain precise sense - the usual approach is to define a formal language, and then require propositions to be expressed in that language. Now, the trick is that (by a theorem of Tarski) you can't set up a formal language in which a proposition can make claims about its own truth. Imagine explaining the sentence "This statement is false" to someone who doesn't quite get it:
YOU: "This statement is false."
THEM: "Wait, which statement?"
YOU: "That one."
THEM: "Which one?"
YOU: "The statement 'This statement is false'."
THEM: "Oh, ok. But which statement is that one talking about?"
And so on. That's basically the problem - a formal system forces you to explain what you're saying so precisely that statements like this just aren't possible.
Now, in practice, it's a pain to put everything in formal language. Generally, what we do is assume - until proven otherwise - that anything that isn't obviously self-referential could be written in formal language if we wanted to. Which is why you see proofs by contradiction of sentences in plain English.
There's a natural next question, though, that you're probably thinking: if we're allowed to limit the "scope" of proof by contradiction by saying "oh, well, it actually only applies to this sort of sentence", then how do we know we don't have to limit it further? The answer is - we don't. Most mathematicians take it as an assumption (an axiom, more or less, called the Law of the Excluded Middle). However, there's a branch of logic - intuitionistic logic - which rejects this assumption, basically saying "Sometimes, sentences might be neither true nor false." This is a perfectly functional way of doing mathematics, though it's typically much harder; like trying to ride a bike with one hand tied behind your back.