Letter of recommendation
All schools will ask you to provide at least two, or more likely three recommendation letters. Letters from non-academic persons don't help, so all have to be written by Professors. Now the question is whom to ask and how to do that. Some ideas, try to pick profs who either studied or worked for some time in the US. There might be a higher chance that admission committees know them, which is always helpful. Otherwise definitely try to get some senior faculty, again the reputation of the author helps. But one word of caution, they should be able to make an informed statement about you. A lukewarm letter from a renowned professor, who doesn't know you, is surely not preferable to a glowing, well-informed, letter from junior faculty. Another problem could be that the professor has no experience with writing such letters, so he might actually ask you to write some draft for him. I did not have to do this, so I don't have experience with it. But as always, the internet has the answer for it, as for almost everything else, there are websites out there (only be sure that you pick one which actually discusses letters relevant for grad school, Business schools and other programs might ask for slightly different things) It might be useful to ask a prof whether he has any advice about getting a Phd, which schools he can recommend, whether he has experience with the admissions process,etc, before you ask him for a letter. This way you can show that you are really interested in his opinion and not just a letter from him. Also provide them with the relevant forms (can be found on school websites, some schools start with electronic forms), but also have other information ready, like your c.v or a list of your courses with grades. Be sure to ask them with enough time to spare, you don't want to pressure them in the end. But take care that they are actually writing it, maybe through little reminder (for example through a question whether they need any additional information). Also be sure that he actually has only nice things to say about you…..
Statement of purpose
This is the part which usually requires the most time and effort. It is also the part differing the most from applicant to applicant. So here are some of my thoughts on it, other people may disagree You can either write a very general SOP or a very specialized one. The second category requires that you know what you want to do, can convincingly make a case for it, show evidence from your previous education and have a reason why the school is exactly the right one to do this. If you write one of these, make sure it is compelling and convincing and that you picked the right schools… Since I didn't know precisely what I want to do, I kept my SOP very general, writing more about my general interest in economics, my fascination with mathematical tools and my reasons for getting a Phd. I included how my previous education prepared me for grad school, what experiences I had with research and how that sparked my interest in doing more. I closed the letter with a paragraph about fields which I am interested in and stating that this school is exactly the right one for this. One discussion is whether you should mention individual professors. My opinion is that it might only be helpful if you write a specialized SOP to show that you investigated the department thoroughly. However since nobody knows whether faculty might actually leave the school, it actually may be counterproductive if you decided to write a more general SOP.
Some general tips:
teaching can be great, but warrants not to be the main reason, it's all about research, research, research
be enthusiastic, the comment I got on my first draft, was something like "you make a good case, but you don't sound enthusiastic enough". So forget European understatement and adapt some American enthusiasm (stopping short of boosting)
don't overstress your successes, yes you might have been top of your class, best here and there, overachiever there, but hey they have your transcripts, so they can see how good you are from there, no need to bore them with rankings and statistics.
Past research is good to mention, but don't get lost in technical details. Try to state that you learned from it, how much you got to love doing research and argue that this is exactly what you want to do.
After they admitted you nobody cares what research interests you wrote in your SOP so only make sure the school actually has someone working in that field.
Have other people read it. Probably under two different aspects, the first regarding the content (like your professors) and the second regarding language (maybe your school has people helping with writing essays and papers)
Keep it short, however the word limits are not exact, so twenty words too much is no big deal, but definitely respect limits.