A sacrifice is giving up something to God, often the killing of an animal.
In the Old Testament, animal sacrifices were repeatedly made to cleanse or purify men of their sins, that is, the things they did wrong.
In the New Testament, Jesus became the ultimate sacrifice, dying once for the sins, or wrong doing, of all people.
The sacrifice of Jesus is covered in all four Gospels: Matthew 27:32-55, Mark 15, Luke 22:63 through 23:55, and John 19. After Jesus died and was buried, he rose from the dead, proving his mastery over death and verifying that through him we can live anew with him in heaven.
The book of Hebrews (chapters 7 through 10, especially Hebrews 10:1-18) connects the Old Testament sacrifices with the sacrifice of Jesus.
In general, an alter is a physical place were sacrifices are made to gods.
In the Old Testament, altars were places were animals were sacrificed to God as offerings or in payment for one’s sins. These sacrifices needed to be repeated again and again because people sinned again and again.
Alternately, altars were sometimes constructed to commemorate an event, agreement, or promise. Here their purpose was more like a monument, used to remind people of what happened.
In the New Testament, Jesus provided an all encompassing sacrifice once and for all for our sins.