It seems unfair after waiting so long for an MBA adcom’s decision that you have to wait some more, but at least you’re still alive. At this point and assuming the school in question is still on your list, there are essentially four things you need to do:
1) Stay calm. A waitlist decision can cause a lot of anxiety ("I'm so close! I need to call them, visit campus, update them about my job, and get an alumni push letter RIGHT NOW!"), but applicants need a clear head and strategy to deal with this new reality.
2) Inform the adcom that you want to remain on the list, usually by email. I strongly suggest you simply tell them yes or no, and don't include a "passion" or "you are my first choice" pitch. The person receiving your email is probably not a decision-maker and they likely don't care. More importantly, this is not the appropriate channel to start your waitlist push.
3) Determine what you can do, if anything, such as send an additional essay or letter of recommendation, provide a test score or career update, etc. Many schools welcome additional materials and updates (based on my current clients' experience, those schools include Tuck, Ross and LBS), while at least one (Wharton) absolutely forbids it. In the latter case, ignore the school's prohibitions at your own peril.
4) If you can send additional information, decide what you are going to send, if anything. I cannot emphasize the following enough: whatever you send or do (including visiting campus) should address the potential weaknesses in your application through new information. Repeating anything that is already in your application is simply a waste of everyone's time. What’s worse is that it may indicate to the adcom an inability to analyze yourself critically.
If your test scores are low, you can try to retake them. If you've reached the GMAT limit, perhaps you can take an online math or English class to demonstrate your awareness of and efforts to overcome your weakness. If your TOEFL/IELTS is low, keep taking it. Meanwhile, outline the classes you are going to take, immersion programs you are going to join, etc. in order to raise your English ability before the program starts. Even better is to have already enrolled in such a program, thereby demonstrating your commitment without waiting for the school's permission or input.
What if you don't have any obvious, numerical deficiencies? Then things become more difficult, because you have to think critically and objectively about all aspects of your application to determine what, if anything, led the adcom to make its decision.
Remember that you might not have any addressable weaknesses and you might not have done anything "wrong". Perhaps the adcom already accepted someone with a similar profile or they thought other applicants were just a touch stronger than you in some respect. In other cases, it might just be bad luck. This being the case, you are not obligated to do anything other than wait.
If you do decide to send "push" materials, first scan your other applications for stories you weren't able to share with your waitlist school that can add to your profile. Other typical items that people include in a push essay:
1) What you experienced and gained from a campus visit after submitting your application.
2) Other efforts to learn about the school since submitting your application (information sessions, alumni meetings, etc.)
3) Promotions or new responsibilities at work, progress on a personal project or endeavor.
4) Progress you've made towards your goals.
Besides these more simple things, ask yourself, "Given my overall application, what would the adcom be most concerned about?" Are my goals both realistic and ambitious? Are there any inconsistencies or contradictions in my essays or other materials? Did I send any negative messages accidentally through my application? Did I leave any information gaps? (These are more subtle issues for which it might be helpful to get an outside opinion.)
A lot of people feel that getting push recommendations from alumni or current students will give them an edge. I tend to discount the impact of such letters since they are just too easy to get. In any case, be sure the alum/current student knows you well and can actually add something new to your profile.
My last piece of advice is to avoid telling the school you are “passionate” for them or that “you are my top choice”. This has become de facto language in push letters and thus means little. The best way to demonstrate your passion and interest is to improve the areas that you can improve as quickly as possible and to then let the adcom know, all while maintaining your professionalism and dignity.