Unless one of the creatures has first strike or double strike, damage will resolve simultaneously. Your creature will deal 4 damge to the opposing creature and the opposing creature will deal 4 damage to your creature. Because the opposing creature has infect, it will deal damage in the form of -1/-1 counters. Your creature will get 4 -1/-1 counters on it, and as a stated based effect of having a toughness equal to or less than zero, will be sent to the graveyard.
I suppose I should add that the opposing creature dies too, having taken 4 damage, enough to destroy it.
Okay, here's what happens in the situation you gave.
4/4 v 4/3 infect.
The 4/4 deals 4 damage to the 4/3 destroying it, while the 4/3 deals 4 damage to the 4/4 killing it. The difference? The 4/3 deals damage in the form of -1/-1 counters so the 4/4 get's 4 -1/-1 counters on it reducing it's toughness to 0, thus putting it into the graveyard. The 4/4 is not technically destroyed because with the 4 -1/-1 counters on it regenerating it will do no good.
In that situation if you gave the 4/4 deathtough the results would be the same. If you gave either of them lifelink, that creatures controller would gain 4 life, whether it was on the infect creature or the non-infect creature.
Now, let's say either of them have first strike. If the 4/4 get's first strike it will deal 4 damage to the 4/3 destroying it BEFORE the 4/3 can put -1/-1 counters on the 4/4. If, however, the 4/3 has first strike it will put 4 -1/-1 counters on the 4/4 before IT has a chance to deal damage to the 4/3 and the 4/4 would be put into the graveyard.
I hope this clears the issue up for you. I only know so much about the mechanics because of wither from Shadowmoor and Eventide.