Disclaimer: my personal insights into this job are personal. Please feel free to engage in the comments if your experience is different.
Alright. Thick Skin and/or Writer Zen. Get some.
Is this new advice? Obviously not. But for pros it's old advice with my perspective and experience added to it, and for those of you just getting into the feel of what it means to be a professional writer it will be new advice.
Listen. You are going to be rejected. Many, many, many times. Often, with nothing more than a form letter. Editors don't have time to pat you on the head and say 'you did a good job but not good enough this time.'
There are plenty of reasons you may be rejected. In the early days of your writing career, it will be because your stories are derivative, ill-constructed and/or boring. As you get some critiquing experience, expand your reading, expose yourself to more writing and hone your skills, you will start getting rejected for many vastly more depressing reasons, and these will be entirely outside of your control. Usually this is because of timing or bad luck.
To give you an example, I recently missed out on an amazing publication opportunity, with a story that fit the criteria perfectly. In the end the piece was rejected because when the final stories in the anthology were collated the genre of my story no longer suited the overal tone of the collection (fyi, mine was more horror than would have been suitable). I only know this because I received a personal email explaining how close to selection it had come, and thanking me for sending the story in. Note: this is almost as cool as actually being published, except without the fame and blackjack and, well, you know the drill.
The lesson here is that any feedback from an editor or reader is good news, and your best story can still not make the cut for reasons entirely unrelated to its quality.
To give you some comparison from my own recent slush-reading experience: if a story is truly woeful I will usually bounce it with a 'No' to the submissions manager. If it really is that bad, offering constructive criticism is too hard given the circumstances. Where do you start? In contrast, if a story has some merit but I still don't want it, I'll usually give some feedback, because as a writer I appreciate feedback, so in my mind other writers ought to, as well. Do as ye would be done by, and all that.
Which leads me to: do not spit or claw at the face that is going out of its way to help you.
Seriously, don't. When you are rejected, suck it up, rage at your friends (not on Twitter or Facebook, editors hang in those spaces, looking for souls to crush). It really sucks, I understand. It's amazing that some editor can't see the value in your submissions. Honestly, I often cry myself to sleep at the injustice of it all, howling defiance into my pillow.*
But...just...stow it. Suck. It. Up. And above all, if you must reply to that rejection email, be gracious, thank them for their time, and let it go.
If you know the story is great, go send it to the next market you've lined up for it; go down the list, submission, rejection, etc, until it sticks. Sometimes a story is actually good, and it's just not right for that particular editor. Usually that's not the case though, so don't rely on that excuse.
If you had some doubts about the story, put it in a (virtual) drawer for a week or two, while you work on something else. Then come back and go over it again, fix it up, and either resubmit it elsewhere or throw it in your reject bin.
Not getting a reputation as a jerk is as useful as not getting a reputation as a hack. Establish a thick skin or (even better) a calm attitude, a writer's zen. Think:
"This too shall pass. Soon, something will be accepted for publication, and if not, I will simply do it again, because there is no alternative. Do, or do not, there is no try."
As a bonus, your writerly composure is a good indicator of your maturity as an artist. Enter your zone or slip on your animal hide outer shell, and you won't be that writer embarrassing their writing colleagues with stories of horrible editors who Just Don't Get Your Genius.
Work on it. And don't let anyone tell you it's easy.
*I don't really cry. I am humouring you.