What happens if you don't use a proxy server?

Is proxy the same as IP address?

I know that proxy is IP address of the computer, but what is IP address?

Is it the same as proxy? You can think of a proxy server as a middleman between you and the remote server. A proxy server typically resides on your local network, so that you can access the remote server without going through a 'big' internet provider, such as Google or Amazon. A proxy server usually consists of one or more web servers (or other servers) that are connected to the internet. The proxy server receives requests from the internet for the IP address of the remote server. The proxy server then goes and fetches that remote server's address. It then returns it back to the client.

If you were on Google, you would access their website via the google proxy server, rather than your own local one. In your case, the proxy server is the one you are using. If you were using Google, the proxy server would be the proxy server at google.

What does proxy do in Wi-Fi?

A proxy server is a program or hardware device which sits in the middle of a network connection.

The proxy listens to both sides of the connection, passing messages between the clients and the servers. Most routers come with built-in proxy capability. So in a sense, you already have a proxy on your wireless router. (The process is more complicated than that, though - the proxy can work via a variety of mechanisms.)
The purpose of using a proxy server is to improve performance and reduce the risk of interception. In Wi-Fi it is common practice to connect into Wi-Fi networks using an external client computer, rather than the network's own modem. This allows the device to spoof its identity - that is, pretend it is connected to a different service provider than it really is. However, some network administrators will see this as insecure and require the device to connect through the Wi-Fi router.

In this case, the proxy is running on the router itself, so that connections which should be going to the public IP address of the router are being sent back to the LAN by the proxy instead. The advantage is that you only need to trust the proxy. The disadvantage is that all the traffic can still be seen and filtered by the administrator, as well as possibly having a negative impact on network performance.

In any case, it's a matter of configuration whether or not a proxy is running on your router. Look for a "web proxy" option under the wireless tab of the router's preferences.