The term Prime refers to any weapon or warframe made with partial or complete implementation of Orokin technology. The weapons and warframes the Tenno use are based on the Orokin's designs; but are not actually Orokin tech. The Prime versions of items are actual pieces of Orokin technology, and frequently sport improved damage, more Polarity slots, or other stat changes that grant them an advantage over their non-Prime derivatives.
Current Tenno technology is derived from the Primes and other Orokin technology, as the Tenno have tried to imitate the success of the Orokin in the form of the non-Prime items. With the discovery of the Voids, many Prime items are being reclaimed by the Tenno and reverse-engineered by Corpus and Grineer, their power being realized once more.
Note that the Seer can be considered prime even though it is the only hybrid weapon in the game thus far.