There are several different things that can affect actual hatch day. The one that most people think about is the average incubation temperature, but humidity, heredity, relative size of eggs, and how they are stored before incubation can affect it. There are probably some others.
I've had eggs hatch two days early, mainly because my incubator was running a little too warm. Many people have had them two or three days late. One common mistake is not counting the days properly. An easy way to remember it is that whatever day you set them is the day of the week they should hatch. If you set them on a Saturday, they should hatch on a Saturday, for example.
What can you expect on hatch day. Loss of sleep. Worry. Extreme anxiety. Lack of interest in anything else. Excitement. Relief. More anxiety.
Oh, you meant as far as the eggs themselves. Some people hear the chicks chirping inside the eggs before they pip. Some do not. Some see the eggs moving around as the chicks position themselves for the hatching process. Some do not. Eventually, you will see a pip. This is where they peck a small hole in the egg shell. Usually, they rest a while after this, sometimes a whole day, sometimes a whole lot less. During this rest, they are absorbing the yolk, the blood vessels in the egg are drying up, and they are preparing themselves for the next step. A lot of people really get nervous during this phase because it often takes so long.
Eventually, they zip. This is where they split the egg shell open. Sometimes they rest a bit after this step, but usually they push the egg shell open and pop out. They are wet and pretty pitiful looking. They will rest again and dry off. After a while, they are up and playing rugby with the unhatched eggs.
I've had hatches where the whole process was over in 18 hours. I've had hatches where the hatch lasted over two days. Hope this helps a bit. Good luck!