MMS issues can be caused by a number things - the issue may not be self apparent.
Here's a fairly definitive guide.
1.) They require mobile data. Your mobile data must be on and enabled even if you are connected to wifi. turn wi-fi off, enable mobile data, try again, make sure you can open web pages. If not - there's your problem.
2.) Is your messaging working at all? Compose a new text to the number 151, write any word, such as "test" and send. Did you get a auto reply? Your SMS messaging is working. No? There's an issue. Contact AT&T tech support. Either services are down, or your account or phone is not set up right.
3.) IMEI mismatch - did you replace the phone, swap sim cards from one phone to another?, Get a warranty or insurance replacement, or otherwise change your phone without contacting AT&T? if you changed your phone, contact AT&T and have them update the IMEI.
Words like " I need to register a new phone for my account" work. When your phone and phone line are properly set up, you should be able to receive MMS texts after doing so, and verifying step 1 - mobile data is working. Different phones require your account to be set up properly. They use different technology types. AT&T must provision or set up the account to match your phone in order for services to work. Swapping a SIM From an iPhone to a Samsung for instance, will result in data and mms issues for the Samsung.
4.) Message center number. Did you go and mess with your messaging settings? If you changed your message center number, and thought it should be your phone number, it's very much not. You won't be able to receive any texts, and your test to 151 above would have failed. There are two possible message center numbers depending on your services. Contact AT&T if you changed the Message center number for the correct one.
6.) SIM Card. More than likely not the problem, but, it can be. Do you have another phone with you that fits the same sim card of the same make and model? if someone else on your account has the exact same phone, swap the sim cards and test. if your SIM works 100% in the other phone, your phone has issues, and may need to be erased and/or replaced. If it doesn't work, you've proved the SIM card may need replacing. Go to a store and get a new SIM, but always contact support first. Just because it works on the other line, that doesn't gaurantee its a SIM problem. A SIM problem will only be determined after AT&T says there are no other problems with your phone settings or account. Don't get hung up on the SIM Card. If you've used mobile data and mms before, or, hours ago.. it's 99.9999% likely not the problem.
5.) An issue with AT&T Network. These things happen. If there is an issue with their network, tech support can either explain it due to an outage, or perform a few actions regarding your account and phone line to attempt to correct the issue. If you have data services working, then likely there isn't an outage and something needs to be reset on their end.
If you can verify your messaging is working, didn't swap or change your phone, data is working, and didn't touch the Message center number, you need to contact Tech support for help, that's all there is to it. Try restarting first. Amazingly that will help a lot. Restarting your phone helps establish a fresh connection to your provider.
6.) No one ever wants to hear this one - your phone is damaged. If all settings on your account are good, you've contacted support, they see no issues with your setup, your SIM Card works, you may have a device issue. At this point, backup your phone, factory reset. Still no? Contact to see if you have insurance. Damage is not covered by warranty.
If the phone is undamaged, you have a limited warranty to replace the phone, if support is able to determine no account issues exsist, you've factory reset the phone, it's not the SIM, etc. etc. a warranty should be fairly hassle free.
That's pretty much all it ever is. Most of these issues come down to a IMEI mismatch (you changed phones without telling AT&T), a data issue (your data is not working) or a problem with their network. If data is working, likely they can reset your network connection and the problem will be corrected upon restarting your phone after.