What is zero zeros? What are no nothings?

It entirely depends on what you mean by 'zero'. Usually in algebra the zero element of some set, with some 'addition' operation has the property that a+0=a where a can be any element of your set. That is, we use addition to define what zero is. If your set of things (affectionately denoted G) also has a 'multiplication' operation, and that operation behaves like multiplication on R then it can be shown that a*0=0 for all a in G. Since 0 is also in G it follows that 0*0=0.