The traditional MBA project is a business plan or a feasibility study. The student selects a business or industry they would like to design a plan for. The business plan encompasses the student's accumulated knowledge and applies it to running and starting a business. The plan features the business' goals, the reasons why they are attainable, financial forecasts, target market, competitive environment, etc. In a feasibility study the student selects a business or venture to invest in and draws up a detailed analysis on that investment proposition.
For MBA students coming from a prior job, a project related to that company/specific industry can have practical effects. The student would select a challenge or issue the company is facing and propose a solution(s) to that problem. The student will do research on determining the causes of the problem and propose viable solutions.
MBA students with a more theoretical bent may choose to take on a more academic project, one that looks at the underlying issues facing businesses. For example, an economics student may choose to examine how a financial crisis impacts global markets. A finance student may choose to look at that same financial crisis and examine how it could have been prevented and propose solutions so that it doesn't happen again. Projects like these may end up being published in trade journals.
Students who are keen on a particular field of study may propose a project that attempts to advance the field. They can chose to do a research paper that may lead to a new paradigm or way of thinking. This can be done by examining current models and comparing them to theoretical models and then testing them. The results of the study may be published in academic journals.