Prime

The term Prime refers to any Weapon, Warframe, Sentinel, Archwing, or accessory made with Orokin technology. The non-Prime weapons and Warframes the Tenno use are based on the Orokin's designs, but are not actual pieces of Orokin technology. Primes frequently sport improved damage, more Polarity slots, or other stat changes that grant them an advantage over their non-Prime derivatives.

The Tenno have tried to imitate the success of Orokin technology in the form of the non-Prime items, resulting in slightly weaker variations. The Tenno were once capable of creating Primes of their own, examples being the Boar, Dakra, and the Kamas, however, such knowledge is now lost. With the rediscovery of the Void, many Prime items are being reclaimed by the Tenno and/or reverse-engineered by the Corpus and the Grineer, the strength of their craftsmanship being realized once more.

Prime Related
While technically not Primes, these units and items implement Orokin technology.

Trivia

 * Using full-model replacement or non-prime skins on items capable of equipping them will completely change the item's appearance to the model used by the skin, ex. equipping the Abra Paris Skin on the Paris Prime will change the weapon's model to that of the normal Paris equipped with the skin.
 * In the case of Prime Warframes, this will only affect the base model of the Warframe, and will not affect any 'attachments' on the Prime Warframe (such as the pauldrons on Rhino Prime). It also doesn't affect the helmet, which is a separate piece and can be swapped out at will from the arsenal for any alternate helmets (including the non-Prime version, regardless of whether or not the player has the non-primed variant).
 * As of you can apply the default skin to a primed Warframe as well as toggle Prime details on or off when using a tennogen skin other than the Prime or default skins.  However, this option is not available for skins such as Immortal skins.