To make it easier to explain, I'll give an example as if all rarities were exclusive to weapons, but they're not. Rarities are global, I just thought it would be simpler to explain if it were so.
NOTE: Runes (whether from Terraria or not), Mystic Symbols, and other forms of enchantment count towards increasing the quality of weapons, even if I don't specify below.
NOTE: (Enemies) examples are just examples and don't take into account the wielder's weapon skills, powers, and spells (or abilities derived from any energy other than mana).
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Gray: Broken, rusty, and almost useless weapons, practically junk, but can still be used in a desperate situation.
Example: Broken piece of a pipe, Rusty and chipped dagger.
Example (Enemies): Rabbits, Foxes, Common Slimes.
NOTE: Very bad prefixes can downgrade a White quality weapon to Gray.
NOTE: Very good prefixes can upgrade a Gray quality weapon to higher qualities.
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White: Normal weapons, made of a not so sturdy and mundane material, or weapons made of better materials but worn out. Can be used in common battles.
Example: Copper sword, Old iron sword. Wand of Sparking (Natural but weak magical items).
Example (Enemies): Wolves (Medium), Big Dogs, Weak magical beasts.
NOTE: Very bad prefixes can downgrade a Blue quality weapon to White.
NOTE: Very good prefixes can upgrade a White quality weapon to higher qualities.
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Blue: Weapons made with good materials and by skilled blacksmiths or craftsmen experienced in their crafts. Can be used in battles against beasts and beings with low mana.
Example: Iron sword, Iron spear, Ice Blade (Before its upgrade).
Example (Enemies): Bears, Tigers, Wolves (Grey), Mother Slime, Medium magical beasts.
NOTE: Very bad prefixes can downgrade a Green quality weapon to Blue.
NOTE: Very good prefixes can upgrade a Blue quality weapon to higher qualities.
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Green: Weapons made with mana-containing materials, or imbued with mana in their creation. Can be used in battles against enemies, beings, and beasts with a moderate amount of mana.
Example: Swords made with magical materials and by a master blacksmith, ancient magic tomes, enchanted bows and arrows, Houtengeki (Before its upgrade).
Example (Enemies): Strong magical beasts, Giant Slimes.
NOTE: Very bad prefixes can downgrade an Orange quality weapon to Green.
NOTE: Very good prefixes can upgrade a Green quality weapon to higher qualities.
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Orange: Weapons made with rare materials and magical forging techniques, weapons of this quality are items created by blacksmiths or craftsmen experienced in their craft, masters in their art and creation.
Examples: Shichishitō (Solar), Houtengeki (After its upgrade), Magic scepters, Swords used by "Heroes".
Example (Enemies): Deerclops (Weakened by imprisonment), Hallucinations (Only Shichishitō), Very strong magical beasts.
NOTE: Light Red quality items cannot be downgraded to Orange.
NOTE: Very good prefixes can upgrade an Orange quality weapon to higher qualities.
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Light Red: Weapons made with something unique or incredibly rare, weapons made by almost legendary masters and craftsmen, enhanced by magical techniques or a great sacrifice.
Examples: Ice Blade (After its upgrade).
Example (Enemies): [REDACTED]
NOTE: Light Red quality items cannot be affected by bad prefixes.
NOTE: Very good prefixes can upgrade a Light Red quality weapon to higher qualities.
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Pink: Weapons capable of killing something beyond.
Examples: [REDACTED]
Example (Enemies): [REDACTED], Demigods, Beings descended from the divine.
NOTE: Pink quality items cannot be affected by bad prefixes.
NOTE: Very good prefixes can upgrade a Pink quality weapon to higher qualities.
NOTE: Weapons of Pink quality and above are exclusive to weapons capable of affecting something beyond.
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Next rarities yet to be unlocked.