Some hearing aids have a feature called a "telecoil" built into them. The telecoil allows the hearing aid to hear magnetic signal representing an audio signal instead (or in addition to) just an audio signal.
A device that is "Hearing Aid Compatible" is designed to output the required magnetic signal that the telecoil can hear in additon to an audio signal.
The term "Hearing Aid Compatible" is usually used to refer to telephones, but may also apply to headphones. Also, hearing aid users who have a telecoil in their hearing aid may use an ALD with a neckloop or with a silhouette to allow them to hear devices that are not by nature "Hearing Aid Compatible".
Headphones: Not all headphones are "Hearing Aid Compatible", but those that use powerful magnets to drive the speakers may be. The most common headphone that people may encounter that is "Hearing Aid Compatible" is the "PhonicEar" Headphone (really an ALD which many movie theaters have to loan free to patrons who would like to hear the movie better. Many users can simply use that ALD acoustically, but users with a telecoil in their hearing aids may benefit from switching it on, since that particular headset is "Hearing Aid Compatible".
Telephones: Most non-portable telephones, some remote phones, and a few cell phones sold in the United States are now "Hearing Aid Compatible". This means that the telephone speaker in the earpiece not only outputs the sound of the person you are talking to, but it also outputs a magnetic signal representing the sound.
All early telephones were automatically hearing aid compatible, because they used magnets to drive the speaker in the earpiece. Telephones built a few years ago were probably not hearing aid compatible, because they frequently didn't use magnets to drive their speakers.
Using a equipped hearing aid with a Hearing Aid Compatible telephone can dramatically improve your ability to hear on the telephone.
Shop carefully, however, since the strength and effectiveness of Hearing Aid Compatible phones and of telecoil varies greatly. If you have more than a minor hearing loss, you may want to check out the several Hearing Aid Compatible telephones and especially those Amplified Telephones designed especially for hard of hearing people.