Mass is a measure of the number of atoms in an object combined with the density of those atoms. Usually people measure mass in kilograms. You can tell how much mass something has by measuring how hard it is to get that thing to change directions or slow down.
If you're driving a big truck, for instance, you're going to need a lot stronger brakes to stop the truck than if you were riding a bicycle. That's because the truck has more mass than the bicycle, and force = mass x acceleration. Even if the truck and the bike were going the same speed, it would take a lot more force to stop the truck, because it has a lot more mass.
Mass is not the same as size - some big things, like balloons, are very light, while some small things, like a lead bullet, are very heavy. The truck we were talking about would be the same size whether it was empty or full, but it would have a lot more mass if it was full - and it would be even harder to stop.
People often get mass mixed up with weight, because when you're on Earth, the two are pretty much the same - things with more mass also have more weight. The truck would be heavier than the bike. But weight changes depending on how much gravity you have pulling on you - the same truck would be lighter on Mars or on the Moon, and it wouldn't weigh anything at all in space, but it would always have the same mass.