Use a period [ . ] at the end of a sentence that makes a statement. There is no space between the last letter and the period. Use one space between the period and the first letter of the next sentence. This goes against the grain for people using the typography instilled by generations of old-fashioned typewriter users, but modern word-processors nicely accommodate the spacing after a period, and double-spacing after a period can only serve to discombobulate the good intentions of one's software.
Use a period at the end of a command.
Use a period at the end of an indirect question.
Use a period with abbreviations:
Notice that when the period ending the abbreviation comes at the end of a sentence, it will also suffice to end the sentence. On the other hand, when an abbreviation ends a question or exclamation, it is appropriate to add a question mark or exclamation mark after the abbreviation-ending period:
Occasionally, a statement will end with a question. When that happens, it is appropriate to end the sentence with a question mark.
Acronyms (abbreviations [usually made up of the first letter from a series of words] which we pronounce as words, not a series of letters) usually do not require periods: NATO, NOW, VISTA, LASER, SCUBA, RADAR. Abbreviations we pronounce by spelling out the letters may or may not use periods and you will have to use a dictionary to be sure: FBI, NAACP, NCAA, U.S.A., U.N.I.C.E.F., etc.