Every cell has a copy of your DNA in it. When a cell divides, it has to replicate the DNA, because one cell turns into two, and each one needs a copy of the DNA. Sometimes mistakes happen during this copying process, and the DNA gets changed slightly – that’s a mutation. When the mutant cell (in other words the cell with the changed DNA) divides, its DNA gets copied so the mutation gets passed on. So in short, a mutation always means that the DNA has been changed in some way.
To answer your other question, yes, cancers are defined firstly by where they arise, so lung cancer always starts from cells in the lungs. But they can be sub-divided depending on what type of lung cell was faulty.