What are the chances that AES-256 encryption is cracked? I'm currently building a web application and would like to encrypt all data on the back-end. I was thinking of using the AES-256 encryption but wasn't sure how safe it was. I did that math and felt safe.

I took this model to a professor at my college, who is a cyber security expert, and he seemed to have quite a different take. He told me there are only two encryption schemes that he knows of that cannot be broken, and AES is not one of them. He said sure, put it in, but just know that there are people out there who can crack it.

I Googled everything I could about the AES and it being cracked, and the only information I could find was this paper: Distinguisher and Related-Key Attack on the Full AES-256
(Extended Version). Sure it says the AES has been cracked but is still has a practical use, right? I believe it is $2^{231}$ instead of $2^{256}$.

So what gives? Is it practically secure to use or not? Does he just know something that most people don't? If my web application database credentials are somehow exposed, and a hacker gets the raw encrypted data, how can I ensure he won't be able to decrypt it within his lifetime?
He told me there are only two encryptions that he knows of that cannot be broken, and AES is not one of them


This tells me you asked your professor the wrong question.  You asked a cyber security expert if an algorithm could be cracked, to which the answer is always yes, with the exception of a handful of inconvenient algorithms, such as One Time Use Pads used in exactly the correct way.  Even in those cases, there's exploits to worry about.

AES-256 is an algorithm.  It can be broken.  If you look at the history of cryptography, every algorithm gets broken eventually.  That's why we make new algorithms.  The question is how long it takes to figure out the math to break it.

The real question to ask is "what is your threat model?"  What sort of attack are you trying to prevent?  Are you creating a digital lock on a diary to keep it safe from your sister's prying eyes, or are you Edward Snowden, on the run from several three letter agencies with billions of dollars of funding?

AES-256 is currently labeled as sufficient to use in the US government for the transmission of TOP SECRET/SCI information.  That's pretty much the highest classification level they could clear it for, so the US government is pretty darn confident that nobody can break AES-256 on the timescales required to protect our nation's greatest secrets.

(Well, almost.  What it really says is that they are confident that nobody outside of the US government can break AES-256.  What you believe that that says about whether the US government, themselves, can break it depends on your threat model... and whether you engage in discussions of whether heavy-duty tin foil  makes better hats or not)