You'll want to find some docs on all the tricks you can do with a where clause, but real quick I think this is what you want:
"SELECT * FROM Scholarships WHERE School = $my_variable";
or
if $my_variable is a text string (notice the useage of "LIKE" and the single quotes around the variable:
"SELECT * FROM Scholarships WHERE School LIKE '$my_variable'";
I personally prefer:
"SELECT * FROM Scholarships WHERE School LIKE '" . $my_variable . "'";