This is an answer from a physicist so take it as such:
Mass is a fundamental quantity and it is defined, in books, as the ability of an object to oppose the action of a force. If you think of pushing (force) a bicycle or a lorry you'll see that the same force will produce a very different acceleration. Simplistically, you can see that the lorry has more "matter" then the bicycle so it will be difficult to move it compared to the bicycle.
Even two objects of the same shape and dimension can have different amounts of "matter" inside (one made of lead and the other of wood, for example) and the only apparent difference between the two will show when you try to push them.
On top of it, mass is always the same and does not change regardless, for example, of the planet you are in whereas the Weight will change (to tell the truth if you are traveling near the speed of light mass changes but this is a kind of extreme).
Hope it helps.