In my household there is an agree-to-disagree truce about how many spaces should be used after a period. It turns out that this argument is intense online, with a lot of incorrect information being thrown around. Surprise, surprise.
I was never taught to use two spaces, so obviously I don't think a second space is necessary. But it turns out that there is a passionate and vocal constituency that believes I am in the wrong. Two spaces after every sentence. Like that.
Is there a right or a wrong in this argument? Despite the dramatic claims of some: no, there's no right or wrong. It turns out that unlike many grammar rules that are firm, the two spaces after a period tradition has more to do with typesetters who long ago used block letters to print text. Today it's considered more of a style choice, if you look at the current Chicago Manual of Style. But they also explain the history a little inaccurately.
Their explanation says that double space use is the legacy of typewriters. Typewriters use monospaced fonts, where each letter is the same width. The manually included extra spaces added a visual break to help the text to be more legible. Even Grammar Girl explains, "For that reason, people who learned to type on a typewriter were taught to put two spaces after a period at the end of a sentence."
But in fact, rules about wider spaces after punctuation marks pre-date typewriters. Even more importantly, what we understand about a space is different from what a typesetter understands about a space. Various space types have names that you've probably never heard, like the em quad, a space the width of a capital 'M' in any font. For a long time the em quad was the standard width to use after a period, which is wider than what you get from a spacebar these days.
Current typesetting also uses many different kinds of spaces, but because everything is highly automated, most of us would never know the difference. There have long been rules to determine what kind of space follows a semicolon, colon, or period, rules that have and will continue to change as technology and style preferences change. If you want to learn more about how typesetters have historically made rules about spaces, I recommend this comprehensively researched article.
It's not wrong to use two spaces, nor is it wrong to use only one. If you like to put in an extra space after each period, nobody should fault you for it, as long as you're consistent. And one could argue that if you're using a monospaced font, such as monospaced Courier, you'd be better served by two spaces after each period, but it's not necessary. You'll find people who are really angry that anyone believes differently from them, but isn't that always the case. Take a deep breath and don't worry about it any longer.