In anime, the idea of different hair colors is somewhat of an inside joke. In Japan, it’s typical for a young lady to start dying her hair. These colors aren’t common to the Japanese, so it makes about as much sense as dying hair pink or green.
Later on, Japanese animators found this useful in portraying multiple characters who would otherwise look too similar to each other. No joke. Plus, black ink was very expensive at the time.
And as hair became the distinguishing feature of characters, artists and animators began projecting characteristics onto hair color. Hair colors usually characterize Final Fantasy characters.
So to illustrate this point, I’m going to draw from these two sources: #1 and #2.
Also, keep in mind that “the brightness of a character’s hair communicates how down-to-earth and otherworldly a character is. The darker a color, the more that character lives in the here and now – and the brighter, the more esoteric, distant, magical and surreal he/she is…

Black: This is the “everyman’s” color (since every Japanese is born with black hair). These characters often see the “big scope” of what’s at stake. When long and flowing, it can sometimes refer to elegance and nobility. Notable types include Noctis, Gentiana, Sazh Katzroy, Barrett Wallace, Zack Fair, Lulu, Auron and Oerba de Fang.

Brown: Brown is also common and safe for main characters, and it usually denotes warmth, friendliness and “normality”. Notable types include Selphie Tilmitt, Yuffie Kisaragi, Bartz Klauser, Noel Kreiss, Tifa Lockhart, Yuffie Kisaragi, Squall Leonhart, Garnet Til Alexandros XVII, Gladiolus Amicitia, and Yuna.

Blonde: Since blonde in many ways is a European hair colour, it stands for foreigners. But more often than not, the protagonist is “special” and pure, dumb or someone who attracts trouble. Notable types include Cloud Strife, Celes Chere, Snow Villiers, Tidus, Rikku, Zell Dinct, Quistis Trepe, Krile Mayer Baldesion, Prompto Argentum and Zidane Tribal.

Orange: These types are usually optimistic and courageous. Notable types include Wakka.

Red: Usually someone hot-headed, head sure, aggressive and rash. Notable types include Oerba de Vanille, Amarant Coral and Ardyn Izunia.

Blue: Usually the cool, soft-spoken, and introverted types. Notable types include Paddra Nsu-Yeul.

Green: Green colors are for the genki (energetic) girls. Most of the time, these characters have dual traits: calming/healing and capability for destruction. Notable types include Terra Branford, Rydia of Myst

Purple: Characters with these colors usually have entitlement issues, and in women are the typical “fantasy princesses”. Notable types include Faris Schweizer, Eiko Carol, and Caius Ballad.

Pink: Pink hair denotes very feminine children. But also characters with this color are in for a great destiny. Notable types include Lenna, Serah, Lumina, and Lightning (in an ironic twist).

White/Silver: This hair color usually refers to someone who is strange, perverse or otherworldly. Notable types include Sephiroth, Hope, Cecil Harvey, and Paine.
Usually, when Final Fantasy needs character development, these characters will have to overcome their inverted traits.