Use quotation marks to indicate that you're quoting someone else, or as scare quotes: i.e., to indicate a sarcastic or skeptical take on a subject. E.g., Yeah, that's some real "blue sky thinking" there.
If you're only quoting a portion of a statement, the quoted material should not be capped, unless proper nouns appear in the quotation. If you're quoting one or more full sentences, use sentence case as normal. (See the example below.)
In most cases, you'll use double quotation marks — " " — but you'll want single marks — ' ' — for nested quotes. E.g., "He told me, 'responsive design is the future.'" In such uses, don't forget that the original double quotation mark still needs to be closed off.
If a quotation comes at the end of a sentence, be sure to place the final punctuation mark inside the quotation marks, not outside. The only exception is when you're using expressive punctuation that wasn't part of the original statement. E.g., He told me "responsive design is the future"!
Quotation marks should not be used with block quotes.