Battle!: You have been challenged to duel, in a fight like this, you can gain power depending on how many hits you land (Academy students don't get stronger by battling, nor do Gennin, Chuunin get 0.1 per hit rounded up, Jounin+ get 0.1 per hit rounded up), you also do not kill eachother, in Battle! you simply fight to show off skills, you also get a Move Limit, which is equal to your HP. When the move limit runs out, you forfeit the match, so Battle! is about strategy more than anything.
Death Battle: in this battle, you have no attack limit, however you do have "limits" your jutsus now have move limits, and you slow down as the fight drags on. In Death Battles, your only objective is to kill the person you're up against (Do not worry, death isn't permanent, but you will be forced into a spirit-like state for a day. Noone can force a Death Battle, however if both players have atleast 1000 HP, and are of different Villages, any "Battle!" will be replaced by a Death Battle, do not be worried though, as you can still choose to spar.
Dodging: Dodging is simple, if you move or jump at the right moment, the opponent's attack will miss you entirely
Missing: Missing is dependant on your stats entirely, calculations have been made to guess at how a battle will go if the opponent is stronger/weaker than you
Equal or generally equal stats: 90% of attacks hit
2X-3X stronger: you never miss, but 80% of their attacks hit
4X-6X stronger: you never miss, but 50% of their attacks hit
7X-9X stronger: you never miss, but 20% of their attacks hit
10X+ Stronger: you never miss, and none of their attacks can hit you