newtonian mechanics - If $F=ma$, how can we experience both gravity and a normal force even though we are not accelerating?

As I sit in my chair, I experience a gravitational force pushing me into the chair and I'm also experiencing the normal force of the chair pushing back at me so I don't fall. According to Newton's Laws, $F=ma$ and I understand that gravitational acceleration near Earth is $-9.8\: \mathrm{m/s^2}$ so the normal force is $9.8\: \mathrm{m/s^2}$ times my mass.

What I don't understand is that if acceleration is change in velocity and my velocity is not changing (thus acceleration is zero), how is there a force?