The problem is that a great many people — including many writers — appear oblivious to this and seem to put complete faith in their spellchecker. The result is that inexcusable typos abound now, even in publications that should know better.*
Here are a few of the most notorious. I haven't included any where the error is actually a widespread misunderstanding about the right word to use (such as lie/lay). These are all mere laziness.
It's/its: One of the two most notorious examples. For the record, it's is short for it is or it has, while its is the possessive form (like my or his) — It's broken loose from its chains. Part of the confusion here is that we tend to expect a possessive to have an apostrophe-s, but remember that other possessive pronouns, such as his, hers or yours, don't have an apostrophe.
There/their/they're: The other of the two. There is used either for a direction or simply to indicate something — There is a man over there. Their is the possessive, just like its — They prepared for their departure. They're is short for they are — They're going to regret their decision to go there.
Lose/loose: This typo is strangely widespread, since the words have not only different meanings but different pronunciations. I'm afraid I'll lose my life. This strap is too loose. Simple. If you find yourself using the wrong one, just stop and think a moment each time you write either.
Bear/bare: One means carry, endure or (with no connection) a large mammal. The other means naked. Someone determined to accept their hardships stoically might talk about how they'd bear everything. If they were intending to bare everything, they might get arrested — at least if they did it in public.
Where/wear/were: Where can be used in various ways, but always indicating place. Where is it? This is where it is. The place where we started. Etc. Wear is either what you do with clothes or it indicates deterioration. The clothes you're wearing show a good deal of wear and tear. Were (which isn't even pronounced the same in most accents) is the past tense plural of the verb to be. Where are the clothes you were wearing?
Hoard/horde: Particularly relevant in fantasy. Dragons have a hoard of treasure; barbarians attack in hordes. A piece of trivia: horde derives from a Turkic word for army, which is also the source of the language name Urdu, originally the lingua franca of the army in India.
Wary/weary: I'm not sure if this is simply a typo, as I've fairly often heard these mixed up in speech. Wary means cautious; weary means tired. There's no connection between them.
Reign/rein/rain: Reign is what a king or queen does; rein is what guides a horse; rain is what falls from the sky whenever you arrange an outdoor event. One case where I often see a confusion here is when people write the phrase giving free reign. It's actually giving free rein — in other words, slackening off the reins to allow the horse to choose its own route.**
Lead/led: This is one that even the BBC's website gets wrong. You lead a horse to water (pronounced leed), but if you did it yesterday, you led a horse to water. However, the metal that's pronounced exactly the same as led is spelt lead. Confused? So are the hordes (not hoards) of people who think the past tense of lead is spelt identically. Incidentally, bear (not bare) in mind that Led Zeppelin may have been one of the greatest bands in the world, but they couldn't spell for toffee.
Effect/affect: There's some justification for getting confused about these words. In the most common meaning, the noun is effect (cause and effect) while the verb is affect (How does this affect the matter?). However, affect can also be a noun meaning an emotion, while it's possible to effect an escape. You really just have to learn these, or else look them up every time.
These are just a few of the multitude of typos your spellchecker won't pick up. The answer? Have a good dictionary*** with you whenever you're writing, whether it's a book or a computer program, and use it if you have any doubt at all.
Not that I'm saying you shouldn't use a spellchecker. As I said at the start, it's a great tool, but it can't do your work for you. A hammer's a great tool as well, but you don't expect it to hammer in nails by itself. A tool is something to be used, not relied on.
* If there are any apparently careless typos in this piece, they're simply deliberate mistakes for you to spot. Honest. Really.
** It's been pointed out that the other spelling actually makes sense, which I don't dispute, but it's not actually the same sense as the original phrase.
*** And I mean a good one. There are plenty of questionable dictionaries out there, especially on line. You're best sticking with an edition of the Oxford Dictionary, although Collins and Chambers also publish good ones. The only good one I know in the US is Merriam Webster, though I'm sure there are others.