There's no built-in method for recording phone calls on your iPhone, but apps which add this capability abound.
Google Voice is a free method for recording calls, but it's limited to inbound calls and requires the other party to dial your Google Voice phone number.
Apps like TapeACall and Call Recorder can record both incoming and outgoing calls to your iPhone, but these apps aren't free.
This may be in part because Apple didn't want to have to deal with the legal issues surrounding recording phone calls - there are laws at both the state and Federal level that place limits on your right to record calls, especially without all parties' consent (and these laws can vary by state).
But it's indeed possible to record calls on an iPhone; you simply need to install an appropriate app to do so.
There are numerous apps available for recording calls, but one offers the ability for free: Google Voice.