I've seen so many answers on Quora that got me lost :
1st answer : The three are similar. Gram atom.... Take the atomic mass of an element and express it in grams (sodium for example atomic mass is 22.990 so 22.990 grams of sodium is a gram atom of sodium. Gram molecule... for this you use the molecular mass of any molecule and express it in grams (water for example two hydrogen atoms plus one oxygen atom atomic mass of 18.015 so 18.015 grams of water is a gram molecule) the gram molecule is also referred to as a mole or mol. Gram ion... virtually the same as gram molecule but referring to ion.
2nd answer : Don't be confused, these are the old names of moles of these things. 1 gram atom means 1 mole of atoms 1 gram molecule means 1 mole of molecules 1 gram ion means 1 mole of ions. Similarly, 2 gram atom means 2 moles of atoms. So, this is the name of mole.
3rd answer : Gram atoms are exactly Total number of atoms present in one mole of a given covalent substance. Gram molecules are total number of molecules in one mole of given substance. Gram ions are total number of ions present in one mole of given ionic substance. Eg - 1 mole of $\ce{NaCl}$ is 1 gram of molecules of $\ce{NaCl}$ and 2 grams of ions of $\ce{NaCl}$. mole of $\ce{MgBr2}$ is 1 gram molecules of $\ce{MgBr2}$ and 3 grams of atoms of $\ce{MgBr2}$.