How is it possible for there to be more than one kind of atom of the same element?

They are called isotopes.

Isotopes have the same number of electrons and protons in their unionized state. They differ in the number of neutrons. The first and simplest example is hydrogen.

The most common hydrogen has

1 proton

1 electron and

0 neutrons

It has 2 cousins

1 proton

1 electron

1 neutron

And

1 proton

1 electron

2 neutrons.

Most elements have some differences in the number of neutrons present in their nuclei. Cesium and Xenon have the most number of isotopes. Each has 36. You wonder how the atoms are held together.

protons determine what element it is, but there are still electrons and neutrons. if, for example, there are three protons in an atom, it's a lithium atom no matter what. however, if it has any more or any less than three electrons, it's an ion (an atom with a charge). if it has any more or any less than three neutrons, it's an isotope (which has a different mass than a normal atom of an element, and is also radioactive).