I cannot tell for sure which one to use with different organizations and enterprises. For example, with a sporting club:
Manchester United's management/administration/board of directors.
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Avoid "board of directors" unless you know you are talking about a real board of directors. In other cases, stick with administration or management.
If you are talking about high-level corporate people who make strategic and executive decisions for a company, organization, or franchise, then management is probably the "safest" word to use in most contexts. Two exceptions to that rule of thumb, though, are government and education. Interestingly enough, Wordnik lists this definition under administration:
The group of people who manage or direct an institution, especially a school or college. [emphasis added]
In a similar way, American Heritage says that the word can refer to:
a. The executive branch of a government.
b. The group of people who manage or direct an institution, especially a school or college.
Oxford lists a more generic definition, but its example usage backs this up:
The people responsible for this [running a business, organization, etc.], regarded collectively : the university administration took their demands seriously
Going back to your Manchester United example:
Manchester United's management decided to...
If you wanted to use a different word for some reason, a better synonym might be:
Manchester United's executives decided to...