@Zophikel Roughly speaking, Philosophy of Mathematics deals with philosophical questions concerning mathematics. One example of a question of Philosophy of Mathematics is "what is number?" (upon which Frege's landmark work Die Grundlagen der Arithmetik is based). For more details on philosophy of mathematics see this.
user131753
On the other hand studies concerning the Foundations of Mathematics can be very roughly said to be the investigation of formal systems and related concepts useful to mathematics via mathematical tools. $\sf{ZFC}$, in this view is just a particular formal system.
@user170039 basically how we preserve mathematics and our beliefs towards what these abstract entities/objects are to us in the real world
user131753
@Zophikel Not always the real world. For example in the extreme form of Formalism we often deny any correspondence of these "abstract" objects to the real world.
