If plugged into an electrical outlet, a laptop will work just fine without the battery. Touching the exposed terminals could result in an electrical shock.
Without a battery, the laptop will not work on its own. It must be connected to a wall outlet in order for it to work. If the charger disconnects, the laptop will shut off.
Laptop owners should not keep their laptop on the charger at all times. The battery needs to naturally drain before being recharged, and this is the only way to keep the battery working for a long time. However, power users should consider replacing their batteries once every 12 to 18 months.