If you drop a glass on a concrete floor it will break. However, if you drop it on a rug, it might not. The speed of the glass as it arrives at the floor is the same in each case. What is different is how long it takes to bring the glass to a stop. Because the concrete floor does not yield at all, the glass comes to a stop very quickly -- perhaps in less than a thousandth of a second. The rug yields a millimeter, giving the glass a little more time to slow down. The acceleration of the glass is the change in velocity divided by the time interval, and so this is smaller for the glass dropped on a rug; and then the Law of Force and Acceleration tells us that the force exerted on the glass is less, too.
We have all seen videos of astronauts floating about in their space capsule, which has led to the widespread misconception that there is no gravity in space. Actually, the gravitational force constant is only slightly smaller for the astronauts. They are only 250 km up, which is pretty small compared to the 6000 km radius of the earth itself. What is really going on is that both the astronauts and the space capsule are accelerating (falling!) towards the earth (because the direction of motion is constantly changing as they move along the curved orbit), and their accelerations are exactly the same -- they are both falling objects (the gravitational force is proportional to mass, and larger masses need larger forces to accelerate). Astronaut and space capsule move along together; if the astronaut is in the center of the capsule, he or she will stay there.
A good example of a situation in which acceleration is very small, consider a teenager who has been asked to take out the trash. Can we apply the Law of Motion to this situation? The mass doesn't seem to be unduly large, and the force that was applied ought to have been sufficient. However, it is the net force that counts in the Law of Motion: evidently there are other forces at work here. Some kind of friction? Or just glued to the TV set?