1. To put it aside for a while. Sounds odd I know but you need to put some distance between you as the writer of the piece before you can become you, the editor of that piece. You are too close to the work to be objective about it just after you've written it. You're either going to think it is the best or worst thing ever written (there seems to be no happy medium here!) so remind yourself, you will look at the piece when it will seem like new to you again. Then and only then can you judge it properly. Assuming you have done that:-
2. Read work out loud. This is great for literally hearing whether your dialogue works as well as you think it does. If you stumble over words or phrases, so will your reader. I've sometimes recorded a story (using Audacity) and played it back. You get to listen to it as a listener would then.
3. Do a basic edit first. I start by getting rid of my known wasted words, repetitions, and go through for spelling and grammatical errors. You will need to do this again at least once more once you've got a final draft but I have found it useful to use this to get me into "editor mode" and to get started on the whole business!
4. Look at whether the structure makes sense. Are there gaps the reader can't follow? Where you have hinted at something happening in the story, did you follow through on it later?
5. Do all of your characters have a vital role in the story? If not, can you get rid of some or amalgamate them into one person?
6. Do all of the plot lines tie up and make sense? Have you shown a point of change in the characters? Have you ensured the story reaches a logical conclusion (which doesn't need to be a happy one)?