Did Jesus really die? I know that Jesus is really God. But I'm still confused... I don't get how God could die or be killed by people. Isn't God always there? How can God die?

Please explain this to me in an easy to understand way.
Sacred Tradition would say yes, the recently reworded Nicene creed says, 


  He suffered death and was buried


We also say it was in:


  accordance with the Scriptures


which means, there's some prophecy that says this had to happen which you can read in Psalm 22 or Isaiah 53 or Genesis 22

Death is the splitting of the body, but not splitting "The Godhead" from the body.  So, no part of Christ ceases to be God upon death. 

The Summa has some other pretty good points concerning this question, I'll summarize:


All humanity was subject to death because of original sin. Jesus was the only one who, because of who He was, could payback this debt through death.
Jesus proves concretely that He did take on human flesh by dying in it.
His death destroys the power of death, knowing God has complete power over death and life gives us comfort.
To set an example of being "Dead to Sin"
So that He could be the first to rise from the dead and give us all hope in the resurrection.


So, it's clear that He did it for a reason, but you're wondering if it is possible.  Some say God can "square a circle" other's don't.  Naive me says God can make a circle square, but "reply to objection 1" me has learned that God cannot deny himself.  He cannot tell a lie and if he says He's going to send us His Son as a Redeemer who will die, then that's what's going to happen.

Furthermore, St. Thomas says:


  Christ did not suffer death which comes of sickness, lest He should seem to die of necessity from exhausted nature: but He endured death inflicted from without, to which He willingly surrendered Himself, that His death might be shown to be a voluntary one. 


Which leaves you hanging, and this is a central point of the Faith, which to wrap around with what I said about the Nicene Creed, is probably why it gets a mention.  If Jesus didn't die, he couldn't have risen and if Christ didn't rise from the dead then:


  And if Christ has not been raised, then empty [too] is our preaching; empty, too, your faith.