Although both versions shown in the title (small capitals, unpunctuated—am, pm—and lowercase, punctuated—a.m., p.m.) are standard, the latter tends to be stylistically preferred and more common. Whichever style you use, use it consistently. Separate the abbreviations from their number with a non-breaking space.
The abbreviations stand for ante meridiem (before noon) and post meridiem (after noon) and are used strictly with twelve-hour time. Since the abbreviations clarify part of day, do not add redundant detail—prefer 11 a.m. or 11 in the morning to 11 a.m. in the morning, though the first is most concise and thus best.
Many people agree that 12 a.m. and 12 p.m. are confusing. Since noon is the meridies, it cannot logically come before or after itself, and midnight, coming both twelve hours before and after noon, should then be both 12 a.m. and 12 p.m. Midnight is also often viewed as the end of day—suggesting p.m.—since conventional habits lead many of us to consider waking as the start of day. (If you find that notion absurd, before its thirtieth edition, the U.S. Government Style Manual recommended 12 a.m. for noon and 12 p.m. for midnight.) Some try to avoid the conundrum by writing 11:59 p.m. and 12:01 a.m., but doing so is clunky. The best solution? In both formal and informal writing, prefer noon and midnight to 12 p.m. and 12 a.m., and when referring to midnight, indicate the date or day as a range. You may thus plan lunch for Wednesday noon and expect submissions by midnight, June 9–10.
With the twenty-four-hour system, if numbers are required, use 00:00 for midnight and 12:00 for noon.