It means ... their max HP. As you damage them their current HP goes down.

do I get to see what the value of either theirs or mine (max hp) is?

Yours, sure. Either in your champ list, or in a battle by pressing on your champ. Theirs, no.

I am not seeing what you mean, in my champ list there are stats for each champ.  Is that HP my MAX HP? If not what is it?

I do not see where pressing on a champ in a battle does anything.

In your champ collection, select a champ, look at the info button that breaks down the champs stats.  There is a 'base' state, in this case hp.  That is the champs base hp before any other factors.  Then it tells you how much is added for gear, masteries, great hall, etc.  the last column adds it all up into total or max hp.

awesome, great explanation

so I assume the max hp is the stat that gets affected by hp reducing effects? 

Personally if they named it Total HP that would be better, imo. At least that would match with some other named label in the game.