Pluto in colourThe goal is to fly to Pluto and take images and measure the atmosphere of the ex-planet and its comparatively large moon Charon. After passing them (the plan is to pass within 10000 km of Pluto) and doing measurements, the probe will continue further into the Kuiper Belt and take measurements of similar objects in the belt.
Now the really cool thing about New Horizons is that Pluto is just incredibly far away. Way beyond comparison to our everyday life. Its distance to the Sun varies between roughly 30 and 50 AU, or 4½ billion to 7½ billion kilometres (1 AU is the distance from the Earth to the Sun - or about 150 million kilometres).
That's a long way. And for that reason it takes the probe 9½ years to get there. You can follow the progress of the probe here, where an image is updated to show you the position of the probe in our solar system (I first saw the graphic here on the Bad Astronomy blog).
The position of New Horizons on Jan 20th, 2010
To get that far it had to go by Jupiter to use the strong gravity as a slingshot to boost the speed of the probe. A bonus from doing that is that it was possible to do various measurements and imaging of Jupiter as well.
The probe recently passed the "halfway" mark to Pluto, in the sense that the distance to Pluto and Earth was the same (a bit over 16AU away from both).
Half the mission time isn't over yet though. The probe was launched on January 19, 2006 and won't arrive at Pluto until July 14, 2015.
It's a long slow walk (not really that slow, New Horizons current speed is 16.44 km/s) to scientific discovery. But it's worth the wait. Flying to Pluto is just that cool :)