The original "object" that is the source of all power in the Cosmere is called Adonalsium. It's not clear exactly what it was or where it came from, but it was eventually discovered by a group of cosmically-aware explorers.
Those people somehow (intentionally) caused Adonalsium to Shatter into 16 pieces, called Shards. A Shard is a piece of Adonalsium that represents the entire manifestation of one of the 16 facets of the original power source. These are named based on their aspect, like Honor or Devotion or Preservation. At any given him, only a single person can possess the power of a Shard.
When a person touches the power of a Shard, but does not keep it, some tiny fraction of that power stays behind in the person, and they become a Sliver. Note that not every person that holds a Shards power is considered a Sliver; for example, Preservation imbued every person on Scadriel with a tiny, almost inert portion if its power, but they aren't Slivers of Preservation. Only Vin, Rashek and Kelsier held Preservation enough to be considered Slivers. The Stormfather on Roshar is also a Sliver; specifically, he was a very powerful and primitive spren who took on part of Honor when that Shard died. The key in each case is that the holder of that power held it long enough that they "kept" some remnant of that Shard's power after it was gone, which continued to affect them (e.g. allowed Kelsier to remain in the cognitive realm much longer than he should have, gave the Stormfather his self-awareness, etc.)
(On a side note: We don't consider Sazed a Sliver because he's still holding both Shards, which makes him a Vessel. If he ever gave up Ruin/Preservation he would become a Sliver.)
A Splinter is the conceptual dual of a Sliver: it's a fraction of a Shard's power that has never been human. We still don't have a clear understanding of what exactly defines a Splinter, or where they come from, but we do know for sure which "things" we've seen so far are Splinters: Seons are Splinters of Devotion, portions of Breath are Splinters of Endowment, some of the spren on Roshar are Splinters of Honor, Cultivation, or Odium. They are self-aware, in the sense that they exist outside of and can act distinct from their Shard. But they are also bound to the "purpose" of the Shard that made them, and appear to have very limited capacity to act counter to that purpose. (See, for example, Sylph's occasional struggle to help Kaladin but being unable to do so until he meets the requirements of Honor).
(Spren are a very interesting showcase of how Shard magic works: spren are a natural phenomenon on Roshar; when the Shards ended up on the planet, each one's magic created it's own new, special types of Spren -- the ones that are self-aware and can bond humans -- that mimicked the primal ones.)
A Shard being destroyed does not seem to have anything directly to do with it being Splintered or not. As far as we know, Endowment is still alive and kicking, and Spren of Honor existed long before Tanavast was killed. We tend to use the term "splintering" a Shard to describe it's destruction, since that is what's in The Letter. But despite the strong similarity in language, there's no direct evidence that splintering a Shard is related in any way to creating Splinters.
Finally: Nightblood is not a Splinter or Sliver of anything; it's just a very very strongly Endowed object.
(Much of this is explained in the very last question from Brandon's Q&A here