What are the benefits of correct tyre pressure?

Improved Vehicle Safety

Under-inflation is the most common cause of tyre failures and may result in a “blow out”, with unexpected loss of vehicle control possibly leading to an accident. Source: Dunlop tyres care & proper inflation

Better Fuel Economy

An American University group have done some calculations to show us how various amounts of psi tyre underinflation have an exponential affect on wasted fuel and additional tyrewear: 
http://www.cmu.edu/cmnews/extra/050921_tire.html

National Tyres and Autocare in UK say "Incorrect tyre pressure can reduce your tyre life by up to 25% and increase your fuel consumption by as much as 5%"
http://www.national.co.uk/information/tyre-pressure.aspx

Victorian Automobile chamber of Commerce (VACC) show why tyre pressure is crucial to car safety.
http://www.vacc.com.au/ConsumerInfo/FivePointSafetyCheck/Tyres/tabid/2036/Default.aspx

Extended Tyre Life

It may not be possible to look at a tyre and actually see under-inflation of 5 psi. However, under-inflation of 5 psi can reduce a tyre’s tread life by 25%. A typical tyre may also lose 1 to 2 psi a month, if not checked and adjusted. Source: Dunlop tyres care & proper inflation

Improve Braking Capacity, Cornering & Vehicle Handling

Properly inflated tyres improve the steering, stopping and traction of your vehicle. Recent studies suggest under inflated tyres can increase stopping distance by close to 8 meters, which could in many cases be the difference between missing and hitting something on the road.

Environmental

Ensuring your tyres are pumped to the recommended level will save up to 100 kilograms of greenhouse gas each year - Department of the Environment

Recently introduced tyre pressure monitoring legislation

The USA National Highway Traffic Safety Administration – (NHTSA) estimates that 49 of 79 deaths and 6,585 to 10,635 injuries could be prevented annually if all vehicles were equipped with tyre pressure monitoring systems.

Following the Tread Act of 2005, NHTSA stipulated that from 2008, all newly manufactured or imported US cars (including all passenger vehicles and light trucks) will be fitted with TPMS.

Similarly in March 2009, the European Union approved legislation measures imposing mandatory adoption of Tyre Pressure monitoring Systems, with an implementation period of two years, starting November 2012.