"Sustain" means to last over time, so sustainability is the ability of something to last. As a civilization, we are concerned about the sustainability of our planet.
Sustainability became a big idea after people began to realize that we were using up our natural resources, so you’re most likely to come across the word when you’re reading about recycling and energy use. If you hear someone talking about sustainability in the context of energy, they’re probably talking about alternatives to oil or coal — both of which will run out in the future. A sustainable source of energy is something that's always there — like water or wind.