I was looking through some apps to store my passwords in on the App Store when I found one that boasted an "AES 256 bit encryption system". Now that sounds all very good and well, but that doesn't mean anything to me if I don't know what it does and what level of security it actually provides. Can anyone give an explanation as to what it is? Thanks in advance.
Typically if encryption is used, it's proper encryption and the fault is in the implementation. Or if the implementation of the cryptosystem is perfect, there could still be a thousand other things wrong. I'm not sure you're asking the right question when asking whether AES is secure.
If you want to judge whether they know what they are doing, you could ask some technical questions like 'which block cipher mode of operation is used?' and 'what kind of MAC-algorithm do you use?', but it's never a guarantee.
To answer your question: AES is perfectly fine. To my knowledge, there are no known vulnerabilities that break the encryption, if it's used properly (i.e. no ECB mode, a proper IV, etc.).
Edit: In the time I wrote this, you got two downvotes. I'm pretty sure that's because people think "you can just google this," and as for AES itself, they are right. It's just not the right question to ask.
Kudos, though, for pausing to think about it! Too many people just glance over the specs when it sounds complicated and assume it's fine. I'm sorry I can't help more.