How can we access the car cigarette lighter power source for other purposes, e.g. as a battery charger or an electric shaver? Does it provide DC power?

The cigarette lighter provides nominal 12 volts DC power (well, actually as high as 13.2V, but everone calls it 12 volts). There are many devices that use this 12 volts to operate - radar detectors, portable GPS units, cell phone chargers etc. You can also buy an inverter that can "step up" the 12 volt DC power to 110 volt AC, suitable for operating a laptop computer or other small appliance. See for example:
Ace - Auto: Car & Truck Accessories: 12 Volt Inverter / Fan / Light: Black & Decker® Power Inverter (PI100AB) (http://www.acehardware.com/product/index.jsp?productId=2151146&cp=2568455.2629363.1258923&parentPage=family)