Quick answer
por qué (two words, accent) = why, in questions.
porque (one word) = because, giving the reason.
el porqué (noun) = the reason.
por que (two words, no accent) = rare, "for which".
Side by side
—¿Por qué llegas tarde? —Porque perdí el tren.Question and answer, and the spelling changes between them.
No sé por qué lo hizo.I do not know why he did it — still a question inside the sentence, so two words with the accent.
No me explicó el porqué.He did not explain the reason — here it is a noun and takes an article.
Memory trick
If you could answer with "because", write por qué. If you are giving that answer, write porque. If you can put <i>el</i> in front, write porqué.
Typical mistakes
No sé porque no vino.
No sé por qué no vino.
An indirect question is still a question, so it keeps the accent and the space. The one-word version would mean "I do not know, because he did not come".
¿Porqué no me llamaste?
¿Por qué no me llamaste?
The noun porqué never opens a question. Direct questions always take the two-word form.
Check yourself
Fill in: "Me pregunto ___ no contesta el teléfono."