The apostrophe (') is a punctuation mark that is used to indicate that a letter or letters have been omitted from a word. It also indicates possession.
When apostrophes are used in place of letters, they create words called contractions. "Don't," "can't," "she'll," and "he's" (do not, cannot, she will, and he is) are just a few examples of contractions.
There are some contractions with two apostrophes, like "wouldn't've" (would not have), but these are generally nonstandard and you shouldn't (should not) use them in writing. Some famous writers, however, like Lewis Carroll, enjoyed contractions with two apostrophes, and would write words like "sha'n't" (shall not).
As aforementioned, apostrophes also indicate possession or ownership. This is likely because, in Old English, people would tack "-es" onto the ends of words to convey ownership. "That is Johnes shoe." As English evolved, people started putting an apostrophe in for the e, and it became "That is John's shoe."
Today, if you wish to indicate that something belongs to a noun, put an apostrophe and an "s" at the end of the noun. Just make sure not to do this when a pronoun owns something!
Pronouns have different rules than nouns. "It is his book." "Is that her shoe?" "I have its toy," NOT "I have it's toy." "It's" is a contraction, a shortened version of "it is." "It's" is not the possessive form of "it." Plenty of people struggle with this one, but just remember: if there's an apostrophe, then that means "it is." If something belongs to it, use "its."
Apostrophes also have uses in quoted material. Read And I Quote to learn about this use of apostrophes.