The apostrophe is that dangling curly thing (or, in some fonts, that dangling straight bit) which all too commonly betrays a novice writer's ignorance rather than helping a story sell. It's also misused in advertising, including yard sale signs and store ads, and young writers may pick up very bad habits that way.
The apostrophe has two proper uses: it signals a contraction (from "do not" to "don't" for example) and it shows possession (John's cow, Ann's saddle, the book's cover, the soldiers' mission.)
Something was needed to show possession because English does not have full declensions anymore (except in pronouns), and the final letter chosen to show both is the same, the "s". One cow, two cows, but also (adding in the apostrophe) the cow's tail, the cows' tails. Languages that still have a genitive form (the understood "of the") don't need an apostrophe or anything else--the word itself changes so you always know when it's plural and when it's possessive. The apostrophe--a small but noticeable mark--was a good choice for this marker.
In one special area, the apostrophe is NOT used to show possession: the possessive pronouns (his, hers, its, ours, theirs, yours) do not use an apostrophe. Pronouns still carry the old case markers. Using an apostrophe with possessive pronouns is wrong. It's not necessary, and misuse creates confusion. Correct :"Its outer casing was broken; we fixed it and it's now working fine."
The apostrophe is NOT used to signal plurals except in a few cases, where the simple "s" or "es" by itself might be confusing. "The following students made all-As" could be read as "all-as" even using a capital letter with lower case, so it is permissible, with all-caps symbols, to use a lower-case "s" with an apostrophe: "The following students made all-A's." However, this limited use does not mean that apostrophes are acceptable (let alone necessary) in all plurals. And if you can avoid it, even in these cases, you'll be doing clear communication a favor.
Misuse of the apostrophe to signal plurals leads to confusion with its legitimate use to signal possession. If you use "cow's" for multiple cows, how are you going to signal that you're talking about the tracks of one cow? Or the tracks of many cows? Already I'm seeing some inexperienced writers reverse the correct usage, and use the apostrophe in plurals and then leave it off in possessives.
Confusing readers this way is not good. Moreover, misuse of the apostrophe is cited often by editors, experienced readers, and successful writers as a sign of poor education and poor writing.
NEVER use an apostrophe in plurals of nouns. If dealing with symbols that cannot be converted to words, consult a good resource or editor. There's often another way. Sometimes just a lower-case "s" or "es" is enough, without the apostrophe.
NEVER use an apostrophe with a possessive pronoun, including "its."
ALWAYS use apostrophes correctly to signal possession: before the "s" for singular, and after it for plurals that end in "s." The old-style plurals that don't end in "s" (such as men, women, etc.) need apostrophe and an "s". (The horse's hoof; the horses' hooves. The man's hat; the men's hats.)
ALWAYS use apostrophes correctly to signal contraction in verbs (put the apostrophe in the right place, and consider the formality/informality of the occasion. Sometimes "do not" is better than "don't" and vice versa. Avoid complex contractions unless it's necessary for a dialect: "shouldn't've" is overdoing it even though we talk like that sometimes. In print, it forces the reader to look twice.)