Four offerings are brought by each of the twelve tribes in the dedication of the altar. But we are not told here what each offering represents. Based on the other passages, I offer this suggestion as to the meaning of each offering:

The grain offering (cf. Leviticus 2) - This offering of flour and oil may represent a thanksgiving offering for God's sovereign providence, by which Adam and Eve was endowed with "every seed-bearing plant on the face of the whole earth and every tree that has fruit with seed in it" for food (Genesis 1:29) and by which Israel would be brought into a land they did not work for, a "land flowing with milk and honey" (Deuteronomy 6:3,10-11).

The burnt offering (cf. Leviticus 1) - As I commented on Matt's sharing below, this offering of blood probably represented atonement (i.e. covering or forgiveness of sin) for innate, constitutional uncleanness, which looks back to the first sin committed by Adam and Eve, by which every human being was thereafter born with a sinful nature (cf. Romans 5:12). Even before the Law was given, both Noah (Genesis 8:20) and Abraham (Genesis 22:2), in their day, offered up burnt offerings to the LORD. 

The sin offering (cf. Leviticus 4) - This offering of blood represented atonement for unintentional sins committed in ignorance, of which David writes "But who can discern their own errors? Forgive my hidden faults" (Psalm 19:12). This offering does not cover willful sins, for which the punishment is either restitution or death. 

The peace (or fellowship) offering (cf. Leviticus 3;7:11-18, 29-32) - This offering of flesh was not for atonement but consumption, whether by God as pleasing aroma (i.e. He savored the smoke from the burning up of the fat and 2 kidneys), or the priests and their families as food (i.e. they received breast and right shoulder), or the people bringing the offering (i.e. they received the remaining meat). The key idea is that every party was pleased, satisfied, contented, gratified with their portion. There was no hoarding for the third day. Everything had to be consumed in the first or second day (Leviticus 7:17-18). This offering takes its name from the Hebrew shalom, which means peace, and so represents the peace that comes when every party involved is satisified.

Thus, in these offerings we have represented (1) thanksgiving for God's sovereign providence, (2) atonement for their sinful condition, (3) atonement for their sins committed in ignorance, and (4) fellowship of peace between satisifed parties. 

Whenever Moses went into the Tabernacle to speak with the LORD, he heard the voice speaking to him from between the two cherubim above the Ark’s cover—the place of atonement—that rests on the Ark of the Covenant. The LORD spoke to him from there (Numbers 7:89).

One of my favorite commentators online, Bob Deffinbaugh, wrote that even Moses could not enter the Holy of Holies in his article on the tabernacle (article, note footnote 144). But he offers no clear support for this assumption. Dr. Deffinbaugh made this observation based on Exodus 40:34-35, which reads,

Then the cloud covered the tent of meeting, and the glory of the LORD filled the tabernacle. Moses could not enter the tent of meeting because the cloud had settled on it, and the glory of the LORD filled the tabernacle.

This text does not say that Moses was never able to enter the Holy of Holies, only that he could not do so on this occasion because of God's glory filling the tabernacle. It is likely that no one was able to enter the tabernacle on this occasion. We have something similar to this in Revelation 15:8, where we read,

And the temple was filled with smoke from the glory of God and from his power, and no one could enter the temple until the seven plagues of the seven angels were completed.

However, when this phenomenon was not occurring, as in our present text, it appears that Moses did go into the Holy of Holies to speak to the LORD. This is also suggested in Exodus 30:6. Furthermore, while the high priest could only go before the LORD once a year and always with a blood sacrifice for himself, Moses had no such requirement (cf. Exodus 25:21-22; 34:36; Numbers 7:89). He could meet with the LORD, it appears, as often as he wanted. But Aaron could not. If Aaron tried to go in before the LORD at any other time then what was prescribed by the LORD, he would die (Leviticus 16:2). Furthermore, while Moses was a Levite, he was not in the line of Aaron, which was the designated high priestly line. Therefore, he did not go in before the LORD as a high priest but as a prophet of God.

What is the profound implications of this? First, the obvious. Moses had a very special relationship with God. This is in keeping with what is described of the LORD's communication with Moses in Exodus 33:11 (cf. Deuteronomy 34:10),

The LORD would speak to Moses face to face, as one speaks to a friend.

In another text, Aaron and Miriam are rebuked by God for speaking badly of Moses. Once again Moses' special relationship with God is highlighted. We read in Numbers 12:8,

With him I speak face to face, 
   clearly and not in riddles; 
   he sees the form of the LORD. 
Why then were you not afraid 
   to speak against my servant Moses?

Notice that in this text God was addressing Aaron and Miriam, who were not just any Israelites. Aaron was the first high priest. But the very fact that Moses could freely go in and out from the LORD's presence meant that Moses ranked above the high priest. 

Through Christ the veil of separation is removed so that believers can go before God, but not as the high priest, who could only go once a year and with a sacrifice, but as Moses. That special relationship Moses had with God is now available to every believer through Jesus Christ. 

Also, what Moses could not accomplish for his people in his day, Christ would accomplish for us. While Moses was declared to be faithful, and people were "baptized into Moses," (1 Corinthians 10:2) they could only enjoy the benefits if they also had the faith of Moses, which most of them did not, for "God was not pleased with most of them, and their bodies were scattered in the wilderness" (1 Corinthians 10:5). This crisis of unbelief is explained in detail in Hebrews 3:1-19. The same could be said for Abraham. Not every offspring of Abraham were saved, only those who were chosen and shared the faith of Abraham (Romans 4:13,16; 9:6-18).

So, those of us that embrace the faith of Abraham and Moses can enjoy the kind of fellowship they had with God through Jesus Christ:

Therefore, since we have a great high priest who has ascended into heaven, Jesus the Son of God, let us hold firmly to the faith we profess... Let us then approach God’s throne of grace with confidence, so that we may receive mercy and find grace to help us in our time of need (Hebrews 4:14,16).

Someone brought up the perceptive question regarding why Moses repeated the offerings that each tribe was to give when they were all the same? Couldn't he have written something like "ditto" for the other tribes after the first. Here was my response to her:

I'm not sure about this; however, it does highlight each tribal leader's obedience to the command. It shows that while Israel is seen as a collective people of God, God also takes note of each tribe, even each individual. Interestingly, they all bring the exact same offering, even though there's nothing to suggest that this is what was prescribed. In other words, the tribal leader of Judah, who was the first to come with his offering, set the standard, and everyone else just followed his lead, no more and no less. As I was reading, I remember hoping that someone would bring two silver plates instead of one or six male goats instead of five. As an application, do we just do what is required? Do we just offer what is expected? What does this suggest about our worship of God?