Step 2 in getting your PS2 online is making sure your PS2 knows how to talk to the internet. That is what the Network Connector is all about.
There are two different kinds of PS2s. There are the original models which did NOT have any network ability. There was literally nowhere to plug a network cable into them. If you have an old, large PS2, you MUST buy a Network Connector. This is a black piece of hardware that screws onto the back of your PS2, and has a network cable port in it. Without this hardware, your PS2 is physically impossible to connect a cable to. There is nowhere to "plug it in".
In the picture below, the rectangle that "bumps out" from the back of the PS2, with the 2 big screws in it, is the network adapter. it is literally screwed into the back of the PS2. You have to buy this separately.
Now, look at the green arrow, pointing at the lower right corner of the network adapter. See the two black ports (holes) there? The lower port is where the network cable plugs into. Again, the port is there because this PS2 has the network adapter on it. The upper port is a phone cable port, because the PS2 network adapter can handle both network cable connections to the internet as well as dial up (via modem) connections.
In addition to that old style PS2, there are now also newer, much thinner PS2s. These come WITH the networking capability built in. Here is a newer style PS2. You can see that the network port is right there already.
So if you have an old style PS2, you must buy a network adapter to go online. There is no other way for your PS2 to have a network cable plugged into it. The USB port will NOT work, that is meant for joysticks only. It has no idea how to deal with a network signal.
If you have a new style PS2, it already has a port in it, and you are ready for the next step.
Step 3 in Getting your PS2 Online