Let's face it: If you're writing, eventually you want to see your work published. Sooner or later you're going to produce something that you think is good enough to submit to a publisher. When you reach this point, whether it's for the first time or the hundredth time, I first want to congratulate you for your accomplishment. Sometimes finishing a draft is the hardest part of writing. I shudder to think how many ideas I've started on, thinking that they would be something great, only to have the idea seemingly "shrivel up" to the point that I don't have anything worth finishing. It's part of the game, and it happens to everyone who tries their hand at writing.
However, finishing a first draft is not the same as having a final copy ready to submit. You always want to edit your work and make sure that it is the absolute best quality it can be. Don't do this immediately after finishing your first draft! Give yourself at least a day before you start the proofreading and editing process (this is why, if a particular market has a deadline, you should try to have your first draft done a few days ahead of that date). When you do open it up to proofread, you're going to have to fight your own familiarity with the story. Make yourself go over it with a fine tooth comb. If you have someone else who can read it with fresh eyes, so much the better.
Things you want to look for that will jump right out at a publishers: Inconsistencies in plot and characterization (i.e., does it all make sense?); outright violations of their submission guidelines (a criticallhy important factor which we will talk about in an upcoming entry); errors in grammar, capitalization, or punctuation; and overly verbose, "clunky" sentences. The latter two are semi-forgivable in that most editors realize that everyone is human, and a limited number of mistakes can be attributed to oversight and fixed without jeopardizing your chances of getting accepted, but you still want to catch as many as you can. However, the first two can constitute a kiss-of-death. Make sure that there is some logic to your piece, and whatever a particular market's guidelines say, make sure you follow that to the letter.