This punctuation mark has two main purposes. It ends all sentences that are not questions or exclamations, and it's used in some abbreviations.
Use periods to break up complicated sentences into two or more readable sentences. "There's not much to be said about the period except that most writers don't reach it soon enough." William Zinsser, On Writing Well: An Informal Guide to Writing Nonfiction, 1980. See sentence length.
Use a period, not a question mark, after an indirect question: He asked what the score was.
Don't put a space between two initials: T.S. Eliot.
Use periods after numbers or letters in listing elements of a summary: 1. Wash the car. 2. Clean the basement. Or: A. Punctuate properly. B. Write simply.
Periods always go inside quotation marks.
Put only one space after a period (and other sentence-ending punctuation, including colons).