The covenant is no small potatoes in Leviticus. It's the salt in the sacrifice. It's the source of everything good. And it's the reason things go way bad.
But what in the world is it?
A covenant is an agreement. In Leviticus, the agreement in question is a suzerain-vassal treaty, normally an alliance entered into by dominant and dependent nations. The use of formal treaty language in Leviticus is one of the ways in which the book tries to move past the magical and supernatural. No ghosts here—this is all about cutting along the dotted line.
As contracts go, this one's a doozy. The covenant defines the relation among God, the Israelites, and the Promised Land. Act consistent with the covenant and the result will be life in abundance: crops, herds, flocks, children, and victory over the community's enemies. But imperfections and mistakes in the agreement can muck up the Tabernacle, the community, and the land itself. The result? Crop failures, thinning herds, and ultimately ejection from the land by the community's enemies or bad directions from Apple Maps.
The contractual relationship also affects all aspects of social life, from family to commerce. In particular, Leviticus regularly describes justice in terms of fair exchange and compensation. Long before the word "redemption" becomes an abstract theological concept, here it refers to buying something back at a fair price. The same goes for atonement, which covers and purifies sin through a ritual form of proportional payment. In Leviticus, payback is not only sweet—it's holy.
This is a big reason why ethical rules can blend into the rituals. For example, the sin and reparation counteract harm to the community from wrongdoing, but they don't rely solely on animal sacrifice. The rules also provide for the offender to compensate injured parties. All in all, this is probably a good thing, because when someone has crashed into your car, you're probably more concerned with the repair bill than whether the person who did it has sacrificed a goat.
While the suzerain-vassal treaty is the big one in Leviticus, it's not the only covenant at play. The burnt offering goes back to the more fundamental covenant between God and humanity through Adam and Noah.
Leviticus makes the link to Adam clear from the outset with the clumsy phrasing in its first rule for a sacrifice, which in Hebrew literally refers to "when a human being brings an offering" (1:2). The word for human being here is adam, and the burnt offering is a direct callback to when God kills an animal to provide Adam and Eve with its pelt as a covering.
The distinctions between clean and unclean animals link to the animals saved by Noah and his covenant with God, which makes the rainbow a sign of God's pledge not to send another flood wipe out humanity. As for asteroids eventually blowing up Russia—well, every contract has a loophole.
God's covenants with Adam, Noah, Israel, and Abraham all come together here: in the rules for ritual purity and holiness.
The ritual purity laws hinge upon such things as covenantal circumcision, the integrity of the covering of the skin, and the leakage of fluids associated with fertility. Gross? Maybe. But it all connects to the covering provided by a house and the physical expression of God's covenantal presence, the Tabernacle.
In chapter 19, the pinnacle of the holiness code, God includes an odd set of commandments pertaining to what Israelites can and cannot eat from a new tree. Now why would a chapter dedicated to God's absolute holiness include a law requiring Israelites not to touch the fruit of a tree? The big hint comes in verse 23, which the King James Version captures in all its literal glorious weirdness:
And when ye shall come into the land, and shall have planted all manner of trees for food, then ye shall count the fruit thereof as uncircumcised: three years shall it be as uncircumcised unto you: it shall not be eaten of. (19:23)
Yep, that's right—the law specifically provides that when an Israelite plants a tree, it is to grow freely for three years "uncircumcised," a rather colorful way of saying untrimmed and unharvested. On a practical level, circumcision means that fruit from year four goes to God (that is, his trusty representatives) and in year five, you get finally get to chow down. (Like this rule? Check out the Tu b'Shevat, the Jewish Arbor Day.)
More generally, this command connects the symbol of God's covenant with Israel through Abraham to the Tree of Life that tempted adam and Eve, the mother of all living. The reference to uncircumcised trees helps explain the connection between male infant circumcision and God's covenantal provision of life in abundance. As it says in the Jewish Mishnah,
Just as the foreskin of trees refers to the place where it yields fruit, the foreskin of man must refer to the place where he yields fruit. (Source.)
Oh, and one last thing. The removal of the foreskin also ties into another theme in Leviticus: imperfection as uncleanness that must be purged. Unlike other covering, the foreskin appears to impede fertility. Once again from the Mishnah:
Just as a bodily orla (foreskin) represents an imperfect situation, and its removal is the correction—i.e., man is created incomplete, and he must perform a certain action in order to bring his body to completion and perfection, so the first fruits are imperfect; they are orla. (Source.)
All of which is to say, the big band song "Don't Sit Under the Apple Tree with Anybody Else But Me" is a part of a long tradition of old-fashioned tree talk that's a lot naughtier than you might think.