Yes, each element has a characteristic number of protons. In other words, each element has a characteristic atomic number. In fact, the periodic table is arranged from left to right by the atomic number. If the atomic number changes, then a new element is formed.
Although the number of protons cannot change for an element, the number of electrons and neutrons can.
Ions are atoms of the same element that have different numbers of electrons. If an atom gains electrons, it becomes negatively charged and is called an anion. If an atom loses electrons, it becomes positively charged and is called a cation.
Isotopes are atoms of the same element that have different numbers of neutrons. Over time, isotopes may be formed by the natural process of radioactive decay.