There are a few laws you should be aware of before you start recording a phone call with your target. There are state and federal wiretapping laws put in place that would limit your ability to do so.
Each state may require consent by one or both parties in a phone call. This is why many of the times you hear the familiar, "this call may be recorded" or asking your permission to record the conversation. If the phone call is taking place in different states, then it gets even more confusing as to which law would over ride the other. In this case, it would be best if all parties consented to the recording of the conversation.
This is a link to the state laws on recording.
As an example, Georgia is a one party consent. Which means that you can record the phone call as long as you are part of the conversation being recorded.
For those of you wanting ideas on what to search for in the app store for both android and iPhone devices. You need to look for ACR or Automatic Call Recorder type apps. There are various free and paid apps available. Some of these may not require the phone to be jailbroken or rooted while others do. For those of you with a rooted or jailbroken phone, there isn't much you can't program the phone to do for you.
Please ensure that you are aware of the law before you start doing your own investigative work. Recording phone calls obviously can be very useful in catching someone admitting they have done something or gaining evidence. I would like to make a note that in order for the conversation to have a solid foundation, it would be important to confirm the identity of the person you are talking too during the conversation, the date and time of the conversation, as well as very specific questions. With a little social engineering, you can achieve this without sounding too obvious.