iPhone and iPad apps tagged "record conversations."
Headset App Makes for an Inexpensive and Handy Alternative Wireless Headset
With the ever-growing list of practical applications for iPhone users that allow your iPhone to become an all-in-one device, sorting out the good from the bad is the only rub.
While you may not have been looking for a way to turn your iPhone into a wireless headset, Shape Services' Headset is an app that does just that.
Manage Voice Memos with Smart Recorder
If the native Voice Memos app doesn't have enough features to satisfy your recording needs, then Smart Recorder may be a good option. There is greater depth, editing options, and a pretty cool method for transferring your recordings to a computer via a Wi-Fi network.