Apostrophes have two functions: to show contractions （I won't） and possession （Shakespeare's tragedy）. They are not used to show the plural of a noun:
    WRONG RIGHT
    Many have criticized the degree to which American's revere personal freedom.
     Many have criticized the degree to which Americans revere personal freedom.
    Machiavelli uses stories of actual prince's to support his argument.
     Machiavelli uses stories of actual princes to support his argument.
    Contractions are acceptable in academic writing, though since some teachers disagree you may have to adapt to different requirements. The main problem apostrophes present for students is in showing possession. The rule is simple, mostly—use an apostrophe followed by s for singular nouns, and an apostrophe alone for plurals:
    WRONG RIGHT
    The Prime Ministers memoirs contain numerous inaccuracies.
     The Prime Minister's memoirs contain numerous inaccuracies.
    For now, the conspirators aspirations survive.
     For now, the conspirators' aspirations survive.
    Some writers use an apostrophe alone for nouns ending in s or z （as I did with species, above）, especially when the next word begins with the same sound. But many teachers will mark this as wrong.
    There is one case where it is preferable to use an apostrophe without an s: for multisyllabic names ending in s or z, especially those of biblical and classical origin:
    Moses' leadership
    Socrates' wisdom
    Jesus' teachings