The “two spaces after period” rule was instituted during the days of typewriters. Typewriters had only one font, so all the letters were monospaced, or took up the same amount of space. That means that the skinny “l” and wider “w” occupied the same amount of space on paper. To make reading easier, the two-space rule was born to give the eyes a break between sentences.
With the dawn of computers, word processing programs not only began offering an absurd number of fonts, but each font was programmed to space characters proportionally (“l” takes up about a third of the space “w” does). In turn, most computer fonts will automatically give you enough room between sentences with one space. So, as a rule of thumb, use just one space when typing up your manuscript on a computer.
There are a couple of exceptions—the fonts Courier and Monaco are still monospaced—but it’s better to stick with one space and switch fonts to Times New Roman or Arial rather than use two spaces.