Gravity
When ignoring air resistance, the forces acting on a bullet are really simple. Since air resistance is removed from the problem that only leaves one force acting on the bullet. That force is gravity. Which gravity is a force that points straight down to the ground and causes a downward acceleration of 9.81 meters per second per second (aka, meters per second squared). However, it is important to note that this is true only when the bullet is no longer in the barell of the gun.
When the bullet is still in the gun barell it will experince a force from the gunpowder exploding and pushing it forward. This is another topic that will not be covered. All that is being looked at today is a bullet that has already left the barell of the gun and is flying through frictionless air.
Some important vector componets to note are the acceleration and velocity vectors. The velocity and acceleration remain constant in the x direction, parrallel to the ground, and in this case acceleration in the x direction is zero. The only thing that is constant in the y direction, perpendicular to the ground, is the acceleration due to gravity and the velocity is always changing. Accounting for these facts, in a frictionless world, a bullet that is dropped and shot at the same time will both hit the ground at the same time, so long as the bullet is shot parrallel to the ground.