What does it mean to say "Jesus died for our sins"? When Christians say "Jesus died for our sins", what do they mean?

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;I went to Catholic school for 5 years and all I heard was "Jesus died for our sins", "Jesus sacrificed himself for us!!", and "Jesus was the Messiah!". What do you mean by this?
As far as I'm concerned, everyone still "sins" all the time.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Which brings me to my next point...

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Why call Jesus a messiah? Was it impossible to be a good person before he "sacrificed himself for us"?

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;This isn't meant to be a hostile question... I really want to know what you mean when you say this kind of stuff.
Jesus died for our sins

What we (and the Bible) mean by the phrase "Jesus died for our sins" is that all sins have a penalty.  We see the same thing in the justice systems of nations--for every crime, there is a penalty.  When the penalty is paid, we say that justice has been served, and that's a good thing.  

Our sins are really rebellion against God, and they carry a weighty penalty.  The penalty is death--not just physical, but spiritual as well.  In physical death, our bodies are separated from our souls and spirits.  In spiritual death, we are separated from God.

Jesus died for our sins in that He paid the penalty for our sins.  As you point out, this does not mean we stop sinning.  What it does mean is that the penalty incurred by our sins is satisfied by the payment made by Jesus Himself.

By way of illustration, if you committed a crime and the penalty for that crime was $1,000,000, you could not pay it.  However, if you happen to be the son of Warren Buffet, he could pay the penalty on your behalf.  The penalty is still paid.  The victim is still restored.  The cost is still real.  Yet because of the generosity and sacrifice of your father, you are not bound to pay the penalty.

So, Jesus died for our sins and paid the penalty for them out of His love and generosity, allowing us to avoid paying the penalty ourselves if we accept that.

Jesus, the Messiah

Jesus is not merely a Messiah--He is the Messiah.  He is not merely one of many good people who made sacrifices for others.  That is not what the word "Messiah" means.  Indeed, many people have done good things and made sacrifices for others, from Mother Theresa to many others.  Yet, while people may make notable sacrifices, no one but Jesus could die for the sins of another.  

All other people are only human and have their own sins to deal with.  Only Jesus, the eternal Son of God who existed before the world and time began, has the authority to die to pay the penalty for the sins of anyone else.

So, Jesus is the Messiah--the only one who could pay the penalty for the sins of the world.