How do I replace my laptop's battery? (Details inside.)

My laptop’s battery is essentially dead. Can’t be charged, can’t retain a charge, at least not for long.

Is replacing the battery as simple as buying a new battery, taking out the old and putting in a new one, as you would do with a flashlight?

Reason I ask is, I seem to recall that in the old days a computer was said to actually require at least a trickle of electricity flowing through the system at all times, so replacing the battery was kind of tricky.

Is this true of today’s laptops?