Ionization reactions for acids. PLEASE HELP!

For this question:

H2SO3: H2SO3 (aq) + H2O (l) => H3O (aq) + HSO3

Why does it remain "HSO3" at the end. I understand why Hydronium forms "H3O" but for "HSO3" why does it remain that. Shouldn't it be "H2SO4"? Its an acid the "H2SO3" I think, but for the ending one "H2SO3" WHY IS IT WRITTEN THAT WAY? I'M SO CONFUSED BY IT.

And then for this question, would it be:

H3PO4: H3PO4 + H2O => H3O + HPO4

AND IF IT IS, THEN WHY WOULD IT BE THAT WAY AT THE END? I'M SO LOST ABOUT THAT!

With acids containing more than one ionizable hydrogen (polyprotic acids), only the first one is pulled of by water. The acids do not ionize unless placed in water:
H2SO3(aq) + H2O(l) <====> H3O+(aq) + HSO3-(aq)
Sulfur forms 2 "oxy acids", H2SO4 and H2SO3, The difference is in the number of oxygen atom attached to the sulfur.
For phosphoric acid the ionization is as follows:
H3PO4(aq) + H2O(l) <====> H3O+(aq) + H2PO4-(aq)
Unless all the hydronium ions are neutralized, the next stage of the ionization will not occur:
H2PO4-(aq) + H2O(l) <====> H3O+(aq) + HPO4=(aq)
The ionization of polyprotic acids occurs stepwise, one at a time. If neutralization is taking place only the first stage takes place in aqueous solutions.

H2SO3, for what it's worth, is called "sulfurous acid", not to be confused with H2SO4 which is "sulfuric acid".

So what you have is the ionization of sulfurous acid into H+ and HSO3-.

If you add this into the original equation, you get;

H2SO3 (aq) + H2O (l) => H3O+ (aq) + HSO3-

Does this make more sense?

Same thing for the second equation, except that it isn't balanced and there are actually *two* reactions that are going on here. I've put them right below, in detail:

H3PO4 + H2O => H3O+ + H2PO4-

H2PO4- + H2) --> H3O+ + HPO4--

When you combine these, your original equation becomes:

H3PO4 + H2O => 2H3O+ + HPO4--