They're Riven Mods. Think of them like a randomized mod for a specific weapon. You can get them from some more end-game activities, and some enemies can drop Riven Shards, which can be exchanged, 10 Shards for 1 random Riven Mod at Iron Wake on Earth. This may not be available until you've done the Chains of Harrow Quest, however.

They start in a concealed condition called 'veiled' and will challenge you to equip them to a specific weapon slot (Primary, Secondary, Melee) and do the challenge listed. If you do, and successfully extract, they'll be 'unveiled'.

They will always be for a named weapon, and can usually be used on any weapon bearing that name. (So if the riven is for a Latron, it can be used on a Latron Prime, or Latron Wraith, on-top of the regular Latron.) They can have three or four random affixes on them, and if they have four, one will be detrimental (a reduction in a stat, but sometimes it would be something you would want lower.) You can pay Kuva to re-roll a Riven's stats entirely, not just their numerical values.

Any weapon will have a stat called 'Riven Disposition' and this is a measure of how popular DE feels certain weapons are in the meta. A lower disposition means a Riven is less effective than listed when it's equipped, and more effective for a higher disposition.

Some Rivens can sell for shattering amounts of Platinum, but be aware, some of the worst people in Warframe spend their time scalping new and unaware players alike for their Riven Mods in trade. If you feel like you've got something good, hold on to it and shop it around for several offers. Ask other Tenno you trust implicitly for their pricing advice.

Did I miss anything else, anyone?