What is the best way to keep your laptop battery healthy?

Keep it in a cool place-Do not let the battery in a high temperature (over 35 degrees Celsius). Ideal in under 25 degrees Celsius.
Unplug when it is fully charged-Do not plug when it is charged fully (do not charge over night).
Plug when it is in low level-Should plug when the battery level is under 15%.

Is it necessary to "calibrate" modern lithium ion laptop batteries?

Battery calibration doesn’t extend its life, rather its purpose is to calibrate its state of charge against the laptop’s “fuel gauge” so the laptop knows when to charge, stop charging, or shut down the laptop when battery charge is depleted - the reason calibration becomes necessary is because the battery’s discharge curve changes as it ages.

It won’t hurt anything to go through the calibration cycle, and doing it will give you the confidence that the fuel gauge will accurately reflect the battery’s state of charge, at least for a couple of years depending on how you use your laptop.

What's the difference between a 4400 mAh and a 4800 mAh laptop battery?

One battery has 400 mAh more, and this affects both how long it will last and the amp output.

Amp output assuming it is a 1C battery then:
4400/1000 = Ah 4.4 (this is to convert mAh to Ah)
4800/1000 = 4.8 Ah

and C*Ah= Amp Output so the numbers remain unchanged and the bigger battery will output 400 mA (.4A) more current

as for how much longer the other will last:
4.4 Ah at 4 amp draw the bigger battery would (theoretically, but voltage drop between the two batteries may make this false) last 6 minutes longer.