A light-year is the distance that light can travel in one year - that is, in the time it takes the planet Earth to make one full trip around the Sun. Light travels 186,000 miles in a second, so in a minute it can go about 11 million miles. In an hour, light travels about 670 million miles. In a day, light travels almost 26 billion miles. So in a year, light travels about 9.5 trillion miles (9,500,000,000,000). That's a lot of zeroes, so we just shorten it and say "one light-year".
Alpha Centauri is the nearest star to the Earth (other than our Sun), and it's about 4.4 light-years away. If we had a spaceship that could travel at the speed of light (which is probably impossible), it would still take almost four and a half years to get there. The whole universe is much, much bigger than that. It's probably about 13.7 billion light-years across.
Because even light can only travel one light-year in a year, when you look at Alpha Centauri in the sky, you're actually seeing what Alpha Centauri looked like four and a half years ago, not what it looks like now. It takes that long for the light to get from Alpha Centauri to your eyes. That means that looking through a telescope at things that are very far away, on the other side of the universe, is the same thing as looking at things that happened billions of years ago. That's how we know what the universe was like when it was first forming, after the Big Bang.