In your case, your probably just need:
cc main.c AddStudent.c
The right thing to do is make a makefile. Here's a (probably a bit naive) example:
myapp: main.o AddStudent.o
cc -o myapp main.o AddStudent.o
main.o: main.c AddStudent.h Headers.h
cc -c -o main.o main.c
AddStudent.o: AddStudent.c AddStudent.h Headers.h
cc -c -o AddStudent.o AddStudent.c
The best place to learn about make is the GNU Make Manual.
Bonus note - if you're starting to learn C, you might want to check out clang. It gives way better error messages than gcc does, in addition to supporting C99 without special flags and being much faster at compiling.
cc main.c -o main -I.BUT function implementations should be left in a C file, so AddStuden.h should actually be AddStudent.cpp since they are implementations of functions from AddStudent.h. In this case, you need to do:cc main.c AddStudent.c -o main -I.