Nostr doesn't have a material existence, it is not a website or an app. Nostr is just a description what kind of messages each computer can send to the others and vice-versa. It's a very simple thing, but the fact that such description exists allows different apps to connect to different servers automatically, without people having to talk behind the scenes or sign contracts or anything like that.
When you use a Nostr client that is what happens, your client will connect to a bunch of servers, called relays, and all these relays will speak the same "language" so your client will be able to publish notes to them all and also download notes from other people.
That's basically what Nostr is: this communication layer between the client you run on your phone or desktop computer and the relay that someone else is running on some server somewhere. There is no central authority dictating who can connect to whom or even anyone who knows for sure where each note is stored.
If you think about it, Nostr is very much like the internet itself: there are millions of websites out there, and basically anyone can run a new one, and there are websites that allow you to store and publish your stuff on them.
The added benefit of Nostr is that this unified "language" that all Nostr clients speak allow them to switch very easily and cleanly between relays. So if one relay decides to ban someone that person can switch to publishing to others relays and their audience will quickly follow them there. Likewise, it becomes much easier for relays to impose any restrictions they want on their users: no relay has to uphold a moral ground of "absolute free speech": each relay can decide to delete notes or ban users for no reason, or even only store notes from a preselected set of people and no one will be entitled to complain about that.
There are some bad things about this design: on Nostr there are no guarantees that relays will have the notes you want to read or that they will store the notes you're sending to them. We can't just assume all relays will have everything — much to the contrary, as Nostr grows more relays will exist and people will tend to publishing to a small set of all the relays, so depending on the decisions each client takes when publishing and when fetching notes, users may see a different set of replies to a note, for example, and be confused.
Another problem with the idea of publishing to multiple servers is that they may be run by all sorts of malicious people that may edit your notes. Since no one wants to see garbage published under their name, Nostr fixes that by requiring notes to have a cryptographic signature. This signature is attached to the note and verified by everybody at all times, which ensures the notes weren't tampered (if any part of the note is changed even by a single character that would cause the signature to become invalid and then the note would be dropped). The fix is perfect, except for the fact that it introduces the requirement that each user must now hold this 63-character code that starts with "nsec1", which they must not reveal to anyone. Although annoying, this requirement brings another benefit: that users can automatically have the same identity in many different contexts and even use their Nostr identity to login to non-Nostr websites easily without having to rely on any third-party.
To conclude: Nostr is like the internet (or the internet of some decades ago): a little chaotic, but very open. It is better than the internet because it is structured and actions can be automated, but, like in the internet itself, nothing is guaranteed to work at all times and users many have to do some manual work from time to time to fix things. Plus, there is the cryptographic key stuff, which is painful, but cool.