How do atoms become isotopes?
By having a different quantity of neutrons in their nuclei.
Let's take the simplest example, atomic hydrogen,
However, a hydrogen nucleus could contain a NEUTRON (a fundamental neutrally charged nucleon). This is still a hydrogen nucleus because the elemental definition goes by
These isotopes are reasonably chemically similar (there are differences but I don't want to make this treatment unnecessarily long and complicated (YEAH, RIGHT!), but they all illustrate the phenomenon of isotopes. Elements (which have the same atomic number by definition) COULD have different masses because their nuclei contain a varying number of NEUTRONS.
Most elements have a few isotopes that are naturally occurring. Chemists can exploit isotopic distribution by the use of isotopic labelling: i.e. doing reactions with different isotopes and seeing where the label (the particular isotope) ends up. Isotopic labels can be interrogated by different means: by mass spectroscopy, where a deuterium label would have the formula mass + 1; or by NMR spectroscopy,
I used the term NUCLEON in this question to describe a nuclear particle (it is an advanced term but is useful in the context). A nucleon is a PROTON or a NEUTRON that comprise atomic nuclei.