Hello, Skjdb,
As the example demonstrates, a cation (with a positive charge, +) is associated with an anion (negative charge, -) in proportions that result in a neutral molecule. Nickel's +3 charge must be offset by three fluorine atoms that have a -1 electrical charge on each. It takes three of them to match Nickel's +3 charge, so we get NiCl3.
If you won an ammonium molecule (NH4 with +1 charge), congratulations and hold your nose. If it's partner were to be a chlorine atom, with a negative charge of -1, they would be happy with one of each: NH4Cl.
For aluminum (Al) and oxygen (O), it is slightly harder. Al has a +3 and O has -2. To get this to be a neutral molecule, we'd need two Al atoms (for a total charge of +6) to mate with 3 O atoms (at -2 each) to reach neutrality. That means Al2O3 would work.
Bob