NAT type 1 means your PS3 is directly connected to the internet through a modem and its IP address is public. Its IP address doesn't need translating so Network Address Translation isn't necessary.
NAT type 2 means your PS3 is connected through a router to the internet and its IP address is part of the router's IP range. Its private network IP address (192.168.0.2, for example) is translated by the router so that incoming traffic destined for it gets routed to it and not any of the other devices attached to the router (like your PC, laptop or smartphone).
NAT type 3 is the same as NAT type 2, only with firewalls restricting its access. This is officially Bad for gaming.
If you're in a house with lots of devices connected to switches or a router, you won't get NAT type 1. Even shoved into the DMZ so that the firewalls don't restrict it and all ports are open, it'll still have a network address that needs translating. For NAT type 1, you'll need a dedicated modem (and line) for just the PS3.
However, type 1 NAT isn't the same as "Open". Type 1/2/3 refers to PS3 NAT, whereas Open/Strict/Moderate refers to game NAT (as reported by certain games) and, usually, deals with port forwarding (or rather Port Address Translation - a subfunction of NAT). A quick summary would be that if the router ports required for the PS3 are shared (by other PS3s, other consoles or other network devices), you'll have a Moderate NAT and if they're blocked or in use by one device you'll have a Strict NAT. If the ports are open and available to the console, it'll be an Open NAT - such as if the PS3 is in the router's DMZ.
So you can have NAT type 1 or 2 and still be Open, Moderate or Strict (whereas Type 3 will probably always be Strict). But plonk the PS3 into the router's DMZ and you'll have NAT type 2 and Open NAT type. Like I do.