If I could make every single job-seeker do one thing, it would be to make them give their application materials to someone else for proofreading before they submit them. Preferably someone with a strong grasp of the English language, but at the very least, someone who's going to provide honest feedback on whether a sentence doesn't make sense or there's a blatant typo.
I've said it a number of times, but it bears repeating: Hiring managers are looking for a reason to cut you from the stack, not a reason to keep you. Maybe it seems unfair that your stellar skills and experience would be rejected because of a small mistake, but remember that you're likely competing against candidates who are just as qualified — and didn't make that mistake.
There are other reasons that surveys of executives show that many will eliminate you for a single typo. Hiring managers assume that you are going to spend more time and energy polishing your resume and cover letter than you will ever spend on a day-to-day assignment at work. If your highest caliber work has errors, then what will your everyday work as an employee look like?
Many positions are looking for someone who is "detail-oriented." Saying you are detail-oriented is more or less meaningless, but letting errors slip by on your applications is a clear way to show that you're not detail-oriented. Again, you don't want to provide an easy way for someone to eliminate you from the long list of possible candidates.
It can be difficult to proofread your own work — your eyes see what you meant to say rather than what's actually there — which is why I recommend having someone else read it over. At the very least, though, read through it once yourself, slowly, possibly out loud, to catch any obvious slip-ups.
When I see a blatant typo on someone's application materials, it makes me think they don't care very much. If they cared, they would have taken the time to go over their materials carefully and had someone else do the same. If it looks like they didn't, then the only conclusions I can draw are that 1) they don't want this job very much, so didn't bother to spend much time on the application, or 2) they're just habitually lazy or careless. Both are good enough reasons to take them out of consideration.
One final note: If you're applying for English-speaking jobs and are not a native English speaker, you don't get a free pass on making errors in your application. Find someone who is a native speaker and get them to proofread your materials.