SHA and AES serve different purposes. SHA is used to generate a hash of data and AES is used to encrypt data.
Here's an example of when an SHA hash is useful to you. Say you wanted to download a DVD ISO image of some Linux distro. This is a large file and sometimes things go wrong - so you want to validate that what you downloaded is correct. What you would do is go to a trusted source (such as the offical distro download point) and they typically have the SHA hash for the ISO image available. You can now generated the comparable SHA hash (using any number of open tools) for your downloaded data. You can now compare the two hashs to make sure they match - which would validate that the image you downloaded is correct. This is especially important if you get the ISO image from an untrusted source (such as a torrent) or if you are having trouble using the ISO and want to check if the image is corrupted.
As you can see in this case the SHA has was used to validate data that was not corrupted. You have every right to see the data in the ISO.
AES, on the other hand, is used to encrypt data, or prevent people from viewing that data with knowing some secret.
AES uses a shared key which means that the same key (or a related key) is used to encrypted the data as is used to decrypt the data. For example if I encrypted an email using AES and I sent that email to you then you and I would both need to know the shared key used to encrypt and decrypt the email. This is different than algorithms that use a public key such PGP or SSL.
If you wanted to put them together you could encrypt a message using AES and then send along an SHA1 hash of the unencrypted message so that when the message was decrypted they were able to validate the data. This is a somewhat contrived example.
If you want to know more about these some Wikipedia search terms (beyond AES and SHA) you want want to try include:
Symmetric-key algorithm (for AES)
Cryptographic hash function (for SHA)
Public-key cryptography (for PGP and SSL)