Some manufacturers integrate a battery protection function in their notebooks. This function only allows charging up to 80% capacity to prolong he lifespan of the battery.
The so-called memory-effect of a Lithium-Ion battery can be disregarded in normal use, a complete charge after a complete discharge is not needed.
On the contrary for many modern devices the opposite is true, it is best to keep the charge between 20% and 80%. This is due to the sensitivity of Lithium-Ion batteries to depth discharge and overcharge. However, in most devices control electronics prevent the user from making any grave mistakes. For example, Apple implements a function which charges the battery as fast as possible up to 80% but charging slows down a lot after that. This conserves the lifespan of the battery at the cost of charge time.
One thing that should be avoided though is leaving your device plugged in for many days in a row. The battery should be discharged once in a while to move the ions inside.
This is a question, that is hard to answer, because it depends on a lot of different factors such as CPU, Display, GPU. Usually a 15.6" notebook consumes about 20W when using the battery. A 17" unit can also use a bit more.
With these 20W a 60Wh battery would last for 3 hours. t = W/P = 60Wh/20W
The energy stored in the battery can be calculated as follows:
W = U * i* t for a 6-cell battery with 4400 mAh this would be 11.1V * 4.4Ah = 48,4 Wh
With the 20W of power consumption this battery would last about 2.5 hours.
If you wish to conserve battery life, the following tips might help you: