For instance, I saw the expression in the Wikipedia article on Lagrange multipliers:
maximize $f(x, y)$
subject to $g(x, y) = c$
What does "subject to" mean?
It is a way to specify constraints. To put it very simply, the problem "do 'X' subject to 'Y'" means that, you have to do "X" (whatever X is), but you have to do it such that "Y" is also satisfied in the process.
As an example, in 1-D
It means that the solution to the optimization problem should satisfy the constraint $g(x,y) = c$.