Watashi is used by females (any age), and males in professional setting (therefore adults). Boku is used by male. Ore（俺/おれ）is another way guys call themselves but used in casual scenes.

私is a gender-neutral pronoun. It can be used by anyone, whatever the age. Luckily, it's a polite-form of the I-pronoun. 僕, however, is a male-only pronoun. You mostly use it among friends, but can also use with seniors who are closer to you. It's bit more rude than 私, but still considered a quite polite pronoun. There are many other forms of I-pronouns too, but most are outright rude, or strictly restricted to professions.