In an ideal world, typos wouldn't be held against you. When your fingers slip and you misspell a character's name, or inadvertently put an apostrophe in the wrong version of its, or forget to close a quotation, it's not as if you've committed some mortal sin. You just let your attention lapse for a moment, and it shouldn't affect your grade at all.
Alas, I've got news for you — this isn't an ideal world. (Heard it here first, folks.) And here's why typos matter: they prejudice your reader against you.
In the pile of papers I'm grading right now — yeah, writing this entry is my way of procrastinating — I've come across one paper that refers to Kate Chopin's work The Storm as The Strom, another that refers to “Zore Neal Hurston” (it should be “Zora”), and another that (in one sentence) confuses conscious with conscience and piece with peace, making for a “conscience decision to revive this peace of mythology.” Now, the rational part of my mind says that the students who made these errors know better; they just slipped, and didn't spot the typos before they printed their papers and handed them in. But when I see a few such errors in one paper, I begin thinking, “This student isn't paying attention to the details. I wonder what else he's missed.” And if that's operating on me at a subconscious level, some poor student who's on the borderline between a B+ and a B might end up with the lower grade because she accidentally wrote that the Declaration of Independence was signed in 1976 instead of 1776.
And in this respect I'm probably re-enacting what would go through the minds of other readers (if, of course, English papers ever got read by anyone besides your professor). If you give the impression that you're not concerned with details, you just hurt yourself in the eyes of your readers. Why should I trust her? She can't even spell medieval, so what are the odds she can teach me anything about the Middle Ages? It's a bad argument, but you're the one who loses by it.
For this reason, if for no other, you should spend a good deal of time proofreading your work, looking carefully for typos. The spelling checker on your word processor can help a bit (although such programs are very far from perfect). Sometimes it helps to put a paper away for at least a day and then to read it with fresh eyes: things you might have missed on Sunday night may be glaringly obvious on Monday morning. But don't let trivial little things like the wrong version of too hurt your chances of getting an A.
Perfection is unattainable: if my own experience is any guide, you can proofread your paper thirty-seven times, and there'll still be an embarrassing blunder on the second line of the first page. (I usually spot my glaring errors about four minutes after I've mailed an essay to my editor.) But know that even little booboos matter, even if your professor doesn't think they should.
[Entry added 18 Dec. 2005.]