In this video I discuss what an FAA TSO is and what it has to do with skydiving.
Every container and reserve must be FAA TSO approved to jump in the US if you're a US citizen or US non-citizen resident. (Foreign visitors can jump non TSO'd equipment.)
There's a misconception that if something is tested to a TSO standard that it's approved, which is not the case. There's lots of gear out there (mostly non-US manufacturers) that have tested to the TSO c23 standard but have not applied nor received FAA approval. Those would not be considered approved equipment.