What exactly is Naruto "Shippuden"?

After certain events in Naruto, there's a two and a half year time skip. The Naruto anime covers the stuff before the time skip, and Naruto Shippuden anime covers the stuff after it. There's only one manga though - the Naruto manga after the time skip is still Naruto.

Simply a portion of the Naruto series.

Much like how Pokemon: BW is simply part of the Pokemon series, how Dragons: Defenders of Berk is simply part of the How to Train Your Dragon series, or how Transformers: Head Masters was simply part of the Transformers series.

As to relationship with the OS (original series), Shippuden is simply set a few years down the road. Content-wise it's also a bit more mature than the OS was.

It is the timeskip where everyone grows older by 3 years. Naruto spent this 3 years training with Jiraiya, Sasuke with Orochimaru and Sakura with Tsunade. Everyone's costume changes. They grew in terms of power too.

It follows Naruto when he is older, basically.