An apostrophe isn't used in place of an s if the s sound is spoken. If you hear an s sound (either unvoiced or voiced) at the end, write an s after the apostrophe.
So, the singular possessive is princess's, the plural nominative is princesses, and the plural possessive is princesses'. All of these are pronounced exactly the same way.
Most nouns ending in an s sound behave this way. For example, the forms for the word bass (the musical instrument) are bass's, basses and basses', all pronounced exactly the same way.
However, irregular plurals form their possessive the same way as singular nouns. For example, the possessives of children and mice are children's and mice's, not children' and mice'.
There's also at least one irregular singular possessive: the possessive of the name Jesus is usually taken to be Jesus' (which is pronounced like Jesus, not like Jesus's) instead of Jesus's.