Phone → Voicemail → Greeting → Custom → Record → Save. So far you probably had a generic voicemail greeting on your iPhone and want to change it to a personal voicemail greeting and with most carriers you don’t even need an additional voicemail recording app and can record a custom voicemail greeting directly on iPhone. Record
It works similar to the old answering machine, with the only difference is that the old answering machine will store the voice message in the machine itself, while your phone’s voicemail system will store the voicemail in your service provider server. You can also forward the voicemail message to someone else’s mailbox.

Forget about messing around with apps. There’s a powerful voicemail handling system built into nearly every big carrier today — and that system almost always includes a system for recording a voice message and sending it directly without ringing the target phone. I will walk you through the process of doing this on Straight Talk (since that happens to be my carrier) but most carriers will have similar systems.
Note: Voicemail, Visual Voicemail, and Voicemail transcription are available from select carriers in select countries or regions.