What makes a 9-cell battery for laptops better than a 6-cell?
Each cell can contain a certain amount of potential energy, based on it's size. Basically, it has a certain number of electrons contained in the package that can be used to power something. The number is going to be based on the chemistry of the battery, but also it's volume. Double the size of the battery, you double the amount of electrons you have to power things.
In the case of 6-cell vs. 9-cell batteries, the presumption is that each "cell" in both batteries is using the same chemistry and physically the same size, so a 9-cell battery takes up 50% more much space / weight but also has 50% more potential energy stored in it - so you get 50% more battery life.
It could be possible to rearrange the batteries to provide a higher voltage instead of extended life. The cells could also be of different sizes and chemistry so that it's not as simple as saying that you get 50% more energy from a 9-cell over a 6-cell (so, it's important to look at the spec sheet for the batteries and compare the mAh ratings of the batteries - these are directly related to how long a battery will last on a charge).
A laptop battery runs on 11.1 volts. That's three 3.7v rechargeable LION batteries.
but I said 6 cell and 9 cell...
Three rechargeable cells in a line. Like a stick. End to end, so that the total output is 11.1 volts. If each battery is a 2600 maH battery, that's 2600 milliamp Hours... then that whole stick will be a 11.1v 2600 maH battery. What's this "maH" or Milliamp Hour thing? Well, that's how many amps of electricity the battery can deliver in an hour. Rather, how many milliamps... and a milliamp is 1/1000th of an amp.
You could say that a 2600 maH battery is a 2.6 aH battery
Now. When you take two of those 11.1v sticks that is made up of three batteries... you get 5200 maH worth of power. 6 cells, two sets of three, would provide twice the power than just one, or last twice as long.
Now... three sticks of batteries...