That's correct, part of grant money is allocated to salaries for graduate students and postdocs, sometimes even salaries for the professor himself. The corollar

That's correct, part of grant money is allocated to salaries for graduate students and postdocs, sometimes even salaries for the professor himself. The corollary is that any PhD program in the sciences where it's the student who pays the tuition is worthless. Another part of a research grant goes to the institution, that's the dreaded "indirect costs".