When you say "push harder" you are talking about force, which makes objects move. The manner in which they move is governed by <PERSON>'s second law, $$F_{net}=ma,$$ where $m$ is the mass of the object and $a$ is the acceleration. Now, in a gravitational field, the weight of an object is equal to it's mass times the gravitational field of the earth, $g$. So the net force is simply $mg$. But if we put that into <PERSON>'s second law, $$F_{net}=mg=ma\rightarrow a=g$$ So, the acceleration is constant, exactly as you expect.
The see-saw works a bit differently. A see-saw is fixed to a point in the center (I'll call that the pivot point), so it will never move (unless you put something way too heavy, and then it will break!). 766
However, the see-saw can still rotate, and rotations are caused by torques. The torque $\tau$ due to a force $F$ acting a distance $r$ from the point of rotation is $$\tau=rF$$ (I am assuming this see-saw is perfectly horizontal, see details below). The further the force is from the pivot, the greater the torque.
In this case, our force is caused by gravity, so each mass is applying a force $mg$ to each end of the see-saw. If you have two masses on either side of a see-saw, they will be balanced if the two torques are equal to each other,
$$r_1m_1g=r_2m_2g$$
So if one mass is smaller $m_1r_2$ in order for the system to remain balanced.
Details: Actually, torque is a bit more complicated. It depends on the angle $\phi$ between the application of the force and the vector from the pivot to the place the force is acting: $$\tau=rF\sin\phi.$$ That's why I made sure the see-saw was horizontal, because the force of gravity will be straight downwards and I don't have to worry about that angle.
EDIT: More details, so everyone understands that these answers are all the same. The actual definition of torque is indeed a cross-product, $\vec{\tau}=\vec{r}\times\vec{F}$, I'm just talking about the magnitude $\tau=rF\sin\phi$. 766