The Nobel Prize was given to Takaaki Kajita and Arthur B. McDonald this year (2015) for their discovery that neutrinos have mass; nearly infinitesimal mass, but some mass, and it matters. Neutrinos are the smallest known particles in the universe. It's hard to conceptualize how small they are though I will try to explain.
The discovery explains some of what we call dark matter. The majority of mass in the universe is "dark" because it cannot be detected like photons; dark matter is detected only by its gravitational influence on objects we can directly detect. Dark matter itself is invisible, hence dark. How much dark matter can be accounted for by neutrinos is unknown.
Neutrinos are important because they probe deeper into matter than any other known particle. We, unlike x-rays, are invisible to any possible neutrino detector. We think we are made of "solid" matter. But we are made of atoms which are 99.999999999999% empty space. What isn't empty space in an atom (outside of its nucleus) is made up of electrons so electrons are much, much smaller than atoms. Neutrinos are less than 1/100,000th the size of electrons; they are so much smaller than atoms that they fly right through them as if they weren't there. For example, by some estimates between 50 trillion and 100 trillion neutrinos originating from the sun's nuclear reactions pass through your body every second (note that the number in the cartoon is way off; more than 65 billion neutrinos pass through your thumbnail each second). They pass through you as if you didn't exist. In fact, most neutrinos from the sun pass through the entire earth as if it weren't there.
However, more than 50 years ago, 100,000 gallons of cleaning fluid buried very deep in a mine in South Dakota produced an average of one argon molecule per day indicating that a single neutron interacted with a chlorine molecule. Raymond Davis, Jr. figured out how to detect the rare argon molecule in the cleaning fluid. Forty years later, in 2002, he was awarded the Nobel Prize for his discovery.
Unlike quarks and other subatomic particles, neutrinos are not constituents of matter. They just fly through the universe so fast that their speed cannot be distinguished from the speed of light. But neutrinos are now known to have mass and therefore cannot, in theory, actually reach the speed of light (there are erroneous reports, likely due to measurement errors, that neutrinos travel faster than the speed of light).
So are neutrinos of any practical use to us? Neutrinos are invisible to us and we are invisible to neutrinos, they don't coalesce into solid matter, and they don't seem to take much notice of matter as they fly through everything in their path. But neutrinos are potentially practical for two important reasons. Neutrinos, unlike anything else we know, can be used to communicate through any amount of matter on earth. One neutrino detector was able to receive an intelligible message through solid rock thicker than a football field. In theory, using neutrino transmitters and detectors, we could transmit messages from Western Australia to Maine through the center of the earth.
A second practical use of neutrinos is that understanding them gets us significantly closer to understanding the universe and the laws of physics that govern it. For example, because neutrinos aren't much obstructed by matter, we can detect neutrinos emerging from an exploding star over 100,000 light-years away. Because neutrinos provide a window into the universe at a far smaller scale than even the Large Hadron Collider in Geneva, the prospects for major breakthroughs in physics are greatly enhanced. The more we understand the universe, the more we can harness natural phenomena to our benefit. This is why the Nobel Prize in physics singled out the scientists who first detected neutrinos for the award this year.