Open the Contacts app (or Phone, then Contacts), choose the person, and tap Edit. Tap Ringtone, pick a tone, and tap Done. That ringtone now plays only when that person calls; everyone else keeps your default. Any custom ringtone you have added to the iPhone appears in the same list.
A different ringtone for each person is where a ringtone that says their name works best: you know who is calling before you look. Ringoz makes one that sings the caller's name, and lets you choose who it is for before you start.
Related questions
How do I make a ringtone with someone's name?
Make a short clip with their name in it, then assign it to their contact.
How do I make a custom ringtone on iPhone?
Get a sound of 30 seconds or less, add it as a ringtone, then pick it in Settings.
What are good personalized ringtone ideas?
Their name plus a line in their voice: a nickname, an inside joke, a catchphrase.