Maxim

A saying is something that is said, notable in one respect or another, "a pithy expression of wisdom or truth."[1]

There are a number of specific types of saying:
Apothegm. “…an edgy, more cynical aphorism; such as, ‘Men are generally more careful of the breed of their horses and dogs than of their children.’" From “What’s the Difference? A Compendium of Commonly Confused and Misused Words” by Jeff Rovin (Ballantine Books, New York, 1994).
Aphorism. A concise definition, notably memorable.
Adage. An aphorism that has gained credibility by virtue of long use.
Chestnut. A long-used and well-known story, comment, or quote.
Cliché. An overly commonplace, hackneyed or trite saying.
Epigram. A poetic form of comment on a particular idea, occurrence, or person.
Epithet. A descriptive word or phrase that has become a popular formulation.
Gnome (Greek: gnome, from gignoskein, to know). A type of saying, especially an aphorism or a maxim, that is designed to provide instruction in a compact form.
Idiom. “…an expression whose meaning can’t be derived simply by hearing it, such as ‘Kick the bucket.’” From “What’s the Difference? A Compendium of Commonly Confused and Misused Words” by Jeff Rovin (Ballantine Books, New York, 1994).
Mantra. A religious or mystical syllable or poetic phrase.
Maxim. A principle or rule. A maxim is a wise saying, especially one intended to advise or recommend a course of conduct. In comparison to its approximate synonyms: saying, adage, saw, motto, epigram, proverb, aphorism, the term maxim stresses the succinct formulation of an ultimate truth, a fundamental principle, or a rule of conduct. The word derives from the Latin word maximus, "greatest", via an expression maxima propositio, "greatest premise".
Motto. A concise expression of motivation used by a group or individual
Platitude. A flat, insipid, trite, or weak remark.
Proverb. An expression of practical truth or wisdom.
Quip. A witty or funny observation.
Saw. A saying that is commonplace, longstanding and occasionally trite.
Witticism. A smart saying, notable for its form or style rather than its content.