What is the difference between average and monoisotopic mass calculations of peptides?
The elements that typically make up peptides and proteins can be present in different isotopes. For example, carbon is present to 98.9% as $\ce{^12C}$, and to 1.1% as $\ce{^13C}$.

The monoisotopic mass assumes that every element is present in the most abundant isotope, the average mass uses the actual distribution of the isotopes in nature.

If you have sufficient resolution in your mass spectrum you can see multiple signals corresponding to different isotope compositions of your molecule.