The first and most important of Israel's national feasts was the Passover. It was first instituted in Exodus chapter 12, when the Israelites were still in Egypt. So here is Exodus 12:1-13 (ESV), which gives the reason for the Passover and how it was to be observed:
The Lord said to Moses and Aaron in the land of Egypt, “This month shall be for you the beginning of months. It shall be the first month of the year for you. Tell all the congregation of Israel that on the tenth day of this month every man shall take a lamb according to their fathers' houses, a lamb for a household. And if the household is too small for a lamb, then he and his nearest neighbor shall take according to the number of persons; according to what each can eat you shall make your count for the lamb. Your lamb shall be without blemish, a male a year old. You may take it from the sheep or from the goats, and you shall keep it until the fourteenth day of this month, when the whole assembly of the congregation of Israel shall kill their lambs at twilight.
"Then they shall take some of the blood and put it on the two doorposts and the lintel of the houses in which they eat it. They shall eat the flesh that night, roasted on the fire; with unleavened bread and bitter herbs they shall eat it. Do not eat any of it raw or boiled in water, but roasted, its head with its legs and its inner parts. And you shall let none of it remain until the morning; anything that remains until the morning you shall burn. In this manner you shall eat it: with your belt fastened, your sandals on your feet, and your staff in your hand. And you shall eat it in haste. It is the Lord's Passover. 1For I will pass through the land of Egypt that night, and I will strike all the firstborn in the land of Egypt, both man and beast; and on all the gods of Egypt I will execute judgments: I am the Lord. The blood shall be a sign for you, on the houses where you are. And when I see the blood, I will pass over you, and no plague will befall you to destroy you, when I strike the land of Egypt.
The film The Prince of Egypt (which is excellent, by the way) depicts the events of the first Passover. On the 14th day of the 1st month, each family among the Israelites killed a lamb at twilight, smeared its blood on their lintels and doorposts, and later ate the meat with unleavened bread and bitter herbs, as God had commanded. On that night, God sent the tenth plague on the Egyptians - the death of their firstborn sons and animals, which caused Pharaoh to release the Israelites. The lamb's blood that was on the Israelites' doors saved them from this plague. God saw the blood and "passed over" them - hence the name Passover.
What was the purpose of this ceremony? Why couldn't God just have passed over the Israelites without the need for sacrificing lambs and putting blood on the doors? There were two main reasons. The Passover feast was to be celebrated every year as a perpetual statute in Israel, to remind the people of how God had brought them out of Egypt into the land He had promised them (Deuteronomy 16:3). This memorial would help the Israelites to love and obey God, as they remembered all He had done for them.
But there was another, even more important reason for the Passover ceremony. In 1 Corinthians 5:7, Paul says that "Christ, our Passover lamb, has been sacrificed." In other words, the Passover pointed forwards to the saving work of Jesus Christ, who sacrificed Himself through dying on the cross in order to make complete atonement for our sins. Just as the lamb's blood on the doorposts saved the Israelites from the plague against Egypt, so Christ's blood, applied to us through faith, washes us clean from sin and saves us from God's judgement against sinners. Christ had to die as a sacrifice because while we are sinners, He is holy and sinless, so God is able to accept His death as payment for our sins. In order to be saved from the penalty for our sin, therefore, we only need to repent and believe in Jesus Christ. That is, we must turn from our sin and trust in Him.
Jesus is the true Passover lamb - so it's no accident that He was crucified during the Passover period. The Last Supper, which Jesus ate with His disciples, was actually a Passover supper. During that meal, Jesus used the unleavened bread as a symbol of His body, and the wine as a symbol of His blood, which was "shed for many for the remission of sins" (Matthew 26:28). So just as the Passover looked forward to Jesus Christ, the communion (or Lord's Supper) which was instituted that night looks back in remembrance of Christ's atoning work. Jesus is at the centre of both ceremonies.