As far as the legal framework goes, it's not a difference of kind, it's just a difference of degree. States had a lot of autonomy, but they had their rights limited by the Federal government in certain areas. That's still the same today, it's just that the Federal government has more power so those limits are broader. Usually, this question is phrased more as a cultural one first rather than a legal one. People viewed themselves as a citizen of a state first and the Federal government was secondary because its reach wasn't as broad. That may have been true, although the federal government's reach was gradual and didn't become what we see today until the New Deal.

A state's right to make the decision to have slavery within their own territory was probably protected, given the understanding of the Constitution at the time. However, pro-slave states wanted more than that. They wanted the right to bring slaves into the territories. I would argue the Constitution is clearly against them on this point. Article IV lets Congress control laws in the territories and it wasn't even novel to let them use this power to ban slavery. The Northwest Ordinance of 1787 banned slavery in that territory. While it predates the Constitution, Article IV's territory clause is plausibly read to give Congress the same power under the Constitution as it had under the Articles of Confederation. In Dred Scott, the Supreme Court decided otherwise, though, which was very controversial. The other thing slave states wanted was the ability to overwrite the laws of other states to allow them to bring slavery into those states temporarily (such as when transporting them to a ship). This is pretty much an explicit weakening of states rights in favor of slave states. The justification is the Full Faith and Credit Clause and the Privileges and Immunities Clause. For this, I'll just say that courts had always rejected this argument. However, a case in New York called Lemon v. People of New York provided an opportunity to rule otherwise. There were people concerned that the Supreme Court would legalize slavery nationally. Regardless, the Civil War interrupted such concerns.

Regarding secession, which wasn't asked, but I'll address, I think it's important to understand the legal theory of states rights (a theory championed by John C. Calhoun and incorporated into Dred Scott). Their theory is that states were absolute sovereign entities who joined together and gave up some of their sovereignty to form a government. Because of that, the government exists to serve the states' interests and, when it fails to do so, the states have the right to withdraw from the entity. While it's plausible to view the Articles of Confederation that way, I don't think the Constitution is best read that way. The founders didn't believe the states were sovereign, they believed the people were sovereign. The people then gave up some of their rights to form a local, state government. Then they gave up some more rights (some of which they previously gave to the state government) to form a national government. It's worth pointing out that state governments didn't ratify the Constitution, instead, special conventions were used so people could ratify them directly. While that doesn't answer whether states have the right to secede, I do think this philosophical justification was simply wrong historically and legally.

We are discussing this in my History university class and our professor tells us that there were arguments for both sides. It basically comes down to how one interprets the constitution. There are traditionalists who feel the constitution should be taken in a literal sense and all powers not expressively given to the federal government are delegated to the states while a more liberal interpretation says that the common welfare clause grants the federal government certain powers to act and superscedes state powers. This division between traditionalists and liberalist regarding the constitution dates back to the ratification of the constitution in 1789 and is one of the early divides between the federalist and the democratic republicans. So your history teacher is right... From a certain point of view. This has been argued back and forth since the constitution was created and there is no definite answer. There were various arguments about whether or not leaving the union was constitutional or not. I know it was a contributing factor to the decline of the Federalist Party when they talked of secession for New England during the war of 1812. It also was a divisionary point between Andrew Jackson and his Vice President John C. Calhoun with the former strictly against it and the other passionately for states rights to both annul and leave the union and their leisure. A cool story is about how once President Jackson and Vice President Calhoun were at a dinner party and they were asked to give toasts. Jacksons famous Union toast was, "The Federal Union: It must be Preserved!" Which Calhoun toast was with The Union: next to our Liberty the most dear: may we all remember that it can only be preserved by respecting the rights of the States, and distributing equally the benefit and burden of the Union!" So as we can see here it was fiercely debated whether or not it was allowed or not.

Edit: The early United States did have the States enjoying more power than they do now. We can even see how this took an effect on language with the United States being referred to with "are" pre civil war while afterwards it was used with "is" For example: The United States are a bastion of freedom for all. The United States is a bastion of freedom for all.

While I hope someone well informed will drop in and clarify matters, the FAQ on the American Civil War would be a good starting point, especially the first part about the causes. From my reading of those threads, it seems that the State's did have the right to regulate slavery, though not to secede.

While those threads don't touch on it, if there was a federal amendment or law approved which regulated slavery, then the stated would have to follow the amendment (being the Constitution), or law, being... a law. Of course, this would require the approval of the slaveholding southern states, making it moot point. The specific bits of the Constitution dealing with the relationship between states and the federal government is Article IV(4). Another rabbit hole you might be interested in diving down is in Article I(1), specifically the "necessary and proper" clause, which gives Congress (the federal government) powers not specifically written out in the Constitution. Again, this is about the extent of my knowledge here and just a starting point to whet your appetite. Hopefully an actual historian will drop by...