1) Read everything on the university website. Everything. Memorize it. Turn off the internet. Write down the material you just read. Tear the paper apart. And write everything again. If you want to get into a good school, you need to know everything about the school, the admission process, all the requirements literally by heart. The LSE website is horrible, everybody knows it, but spending 20-30 hours browsing and getting acquainted with the system is very useful. Many people neglect this advice and either contact a current student or (much worse) a university officer with regards to a question which is readily available on the website (e.g.: program deadlines, required documents, etc). Do not make yourself look like an idiot. Also, research all the programs available and make sure you apply for areas in which you are actually interested in. The worst case scenario is to find yourself in the middle of December of your course year thinking that you are studying something you really, really do not like. So, make sure you know what you are getting into before hand.
2) Read admissions forums. I personally had an account on the Studentroom and a couple of other niche places. Never underestimate the intelligence of other applicants (both current and past) and some of the advice I got from the forums I then applied to my own application.
3) Diversify. I know that everyone wants to be the top dog but, just like everything else in life, you don't always get admitted to your target school. Applying to 8-10 programs is usually considered optimal. Some people make as many as 20 applications. I applied to 6 programs in the UK, the Netherlands, and Spain. Don't apply just for the sake of applying. Be careful and do not, please for God's sake, do not make grammatical mistakes in your applications.
4) Never underestimate the GRE/GMAT scores. People often get in love with their grades, which is a dead-end strategy. Every good applicant was number 1, had perfect grades since kindergarten, is in the top 1% in the universe according to his parents and his ego. The universities do not usually care about your grades if you are an international student from outside Europe and the US (or a top school in Asia). So for someone applying from Azerbaijan, GRE/GMAT scores are basically the only way to demonstrate superior academic intelligence. Getting a 700+ on GMAT or 90% on the GRE quantitative is necessary. It is not sufficient for getting into a good program but if you get this threshold you can be comfortable that you will _never_ get rejected because of poor standardized test results. With regards to English knowledge, everybody knows English. If you are proud of your high TOEFL or IELTS result then this is very sad - it just means you know how to speak. But so do the best applicants from the UK. So, why should the university admit YOU and not a native speaker? Passing the minimum TOEFL/IELTS requirement does not help at all; but failing it is often disastrous. Basically, just get a 100+ on the TOEFL ibt. I am not familiar with IELTS very much.
5) Grades don't matter as much as you think. Otherwise, every top program would be populated by high performing students from mathematically oriented majors such as engineering or pure math. But it is not always the case. Some of my classmates majored in history, philosophy, geography. They were top dogs in their respective areas but very few actually were valedictorians. Of course, excellent grades indicate intelligence to some extent, but it is never ever just about getting all As. It is more about the complexity of the courses you have taken and whether you were brave enough to take on challenging courses. Naturally, mathematically oriented courses are hugely advantageous; but usually nothing above standard multivariate calculus, linear algebra, and statistics/probability is needed for a decent graduate program. For PhD admissions, it is a bit different since most successful applicants (I am speaking of economics PhD programs, the good ones) have good grades in real analysis, basic topology, differential equations, dynamic programming, stochastic calculus, etc.
6) Letters of recommendation are everything. To be honest, I got into LSE because I had good referees. Of course, to get good letters from good referees is quite hard - you need to impress them, work very hard, but if an established professional academic in your field can comment on your skills and potential and say something along the lines of "this guy is in the top 5% of anyone I have ever taught" - trust me you will get admitted to most places. And you can ask anyone else in any other good program (especially economics) why they ended up where they ended up - their response has to include good reference letters. One outstanding recommendation can land you at Harvard, and one bad one can generate 15/15 rejections. If your reference letter is from a foreign-based professor, preferably with a PhD from an American or good European school - this is fantastic. If you don't have a reference from someone residing abroad, then make sure your references are from people who received their Masters/PhD from good foreign universities. Please, make sure they can write in normal English... So many of these references have mistakes. If your referee can't write - then really I can't help you with anything else. Getting a letter from somebody who has never done serious research or has never been good enough to get admitted to a top school herself will never be valued as highly as a recommendation from a "solid" guy. He doesn't have to be a Nobel laureate; in fact, the younger professors usually write longer letters. Just make sure you diversify here as well - 2 letter from "top" professors (including your thesis advisor) and one outstanding letter from a younger lecturer who likes you. Just trust me on here - recommendations will define your life. Focus all your might on getting good recommendations. Sometimes, it is worth spending 1 year doing a teaching/research assistantship for a tough professor just in return for a 1 paragraph recommendation. But you have no idea how much magic that 1 paragraph will do to your admission chances. How to get a good referee to write for you? Teaching or research assistantship, outstanding work on a dissertation that the referee supervised, and demonstrated interest in doing graduate work (go to the referee's office hours several times and convince them that you are ready and committed)
7) Application essays. These are one of those things that rarely, if ever, make a huge positive difference if other parts of your application are already bad. In other words, if your test scores and grades are fine, recommendations are from good professors, and the essays are average - then you are in pretty good shape. But mistakes in the essay, or writing something completely irrelevant or downright stupid - will have a negative effect. Just come up with a somewhat original, structured story about how you have developed an interest in the field, any work experience, how you interacted with the supervisor and what inspired you to study graduate economics or anything else. Do not be cliche and do not overdo creativity. Please, do not try to be humorous and make jokes. Stay professional, concise, but interesting. It is much better to simply "go through the motions" and write a standard essay and focus on more important parts of the application (tests and references) than taking your chance with an ultra unorthodox essay. In my case, my essay for the LSE had 1 grammatical error in it, for what it's worth. But overall I put on about 2 weeks of working on the essay basically every day for 4 hours (apart from other duties I had), polishing out the phrasing and style.
8) Work experience. It helps. Really, the universities claim that they never prioritize applicants with work experience, but this is not true. I am not saying you should work for 2 years before getting a Masters degree (I did), but at least 1-2 summer internships at a firm/bank/organization where you did real analytical work such as data analysis, audit, financial statement preparation, or something numerical - this will make you stand out a bit. Again, for MBA and MPA degrees you definitely need relevant commercial or government experience. If you are shooting for a research degree, then research experience at your central bank, ministry of finance, IMF, World Bank, or just your university research center (or your supervisor) - will be a fantastic addition to your package. Volunteering also always looks very good on your CV. Teaching experience also sends a signal, although it depends on where you are applying.
9) University fairs and recruitment events. If you have the financial capacity to attend some sort of Open Day and meet the admission officers (or maybe even professors) responsible for your program, then it will definitely send a good signal about your serious interest. But I have not attended any such event for any university that admitted me. I am sure 95% of my classmates didn't. Now, getting a summer school course at the LSE before applying for their masters program is an amazing strategy. Say, if you are applying next Fall, then apply now for their Summer program in Summer 2014, take a difficult course (Introduction to Econometrics, for example), get a good grade there, even impress a professor and ask him for a recommendation. I didn't do this because I didn't have the money for the expensive LSE summer school. But this definitely is a powerful move.
10) Additional things. Not really needed. Do not waste your intellectual effort on too many fronts. Just focus on 5-6 things but do them very well: solid grades, necessary test score results, very good recommendation letters, some work experience, demonstrated commitment to the new journey, and a bit of luck. Luck, unfortunately, does play a role. But if you apply to a large and diversified enough pool of programs - you will get admitted for certain.