Explain why atoms have different isotopes. In other words, how is it that helium can exist in three different forms?

The number of protons, massive, positively charged nuclear particles, gives #Z#, the atomic number, which defines the identity of the nucleus.

#Z=1, "hydrogen",Z=2, "helium",Z=3, "lithium",...........Z=21, "scandium".#

The nucleus may contain various numbers of neutrons, massive, neutrally charged nuclear particles, which gives to the existence of isotopes.

For hydrogen, #Z=1# by defintion. That is there is a proton in all hydrogen nuclei. Some hydrogen nuclei contain 1 neutron to give the deuterium isotope, #""^2H#; an even smaller number of hydrogen nuclei contain 2 neutrons, to give the tritium isotope, #""^3H#; Most atoms have a number of isotopes in different proportions.

To answer your question, the helium atom has #Z=2#; i.e. there are 2 nuclear protons, which is what defines the atom as helium. The most common isotope of helium is #""^4He#. #""^3He# and #""^5He# are known with vanishingly small abundances, and half lives.

This video discusses how to determine the number of subatomic particles in C-12 , C-14 , H-1 , H-2 and H-3 .