How to send information from a web-site to the arduino?
Yes of course!
First things first - Yes, it is possible to accomplish what you're thinking.
The idea is to create a server on the Ethernet Shield and open the webpage hosted by it on any device and based on the options selected on the web page we can trigger desired actions on the Arduino.
I've modified the web server code of Arduino built-in examples to match your requirements, just upload it and make necessary changes as required, here it is-
#include&lt;SPI.h&gt;
#include&lt;Ethernet.h&gt;

byte mac[] = {
   0x00, 0xAA, 0xBB, 0xCC, 0xDE, 0x02
 };


 IPAddress ip(192,168,1,199); // Any ip address not on your network.
 EthernetServer server(80);   // Port 80 is default for http.

  int led=2;

  String request;



  void setup(){
    Ethernet.begin(mac,ip);
    pinMode(led,OUTPUT);
    Serial.begin(9600);
   }



  void loop(){

      EthernetClient client = server.available();
  if(client)
         {
       Serial.println(&quot;Client is trying to connect......&quot;);
       while(client.connected())
       {
  if(client.available())
  {
    char c = client.read();
    request.concat(c);
    if(c=='\n'){
    Serial.print(request);
    if(request.indexOf(&quot;ON&quot;)&gt;0){
      digitalWrite(led,HIGH);
      Serial.println(&quot;ON&quot;);
      }

      else if(request.indexOf(&quot;OFF&quot;)&gt;0){
        digitalWrite(led,LOW);
        Serial.println(&quot;OFF&quot;);
        }
    client.println(&quot;HTTP/1.1 200 OK&quot;);
    client.println(&quot;Content-Type: text/html&quot;);
    //client.println(&quot;Refresh: 5&quot;);  // refresh the page automatically every 5 sec
    client.println();

    client.println(&quot;&lt;!doctype html&gt;&quot;);
    client.println(&quot;&lt;html&gt;&quot;);
    client.println(&quot;&lt;head&gt;&lt;title&gt;LED&lt;/title&gt;&lt;/head&gt;&quot;);
    client.print(&quot;&lt;!DOCTYPE HTML&gt;&lt;html&gt;&lt;body &gt;&lt;a href ='/?LED_ON'&gt;&lt;p style = ' text-align:center'&gt;LED ON&lt;/p&gt;&lt;/a&gt;&lt;br&gt;&lt;a href ='/?LED_OFF'&gt;&lt;p style = ' text-align:center'&gt;LED OFF&lt;/P&gt;&lt;/a&gt;&lt;br&gt;&lt;/body&gt;&lt;/html&gt;&quot; );
    client.println(&quot;&lt;/html&gt;&quot;);
    client.stop();
    request=&quot;&quot;;
    
  }
  }
 }
  
    }


   }

This sketch will turn an LED on or off attached to digital pin 2.
You have to go to the IP address that you provide to the shield (192.168.1.199 in my case). You'll see 2 hyperlinks to switch on or off the led and that would pass a query which the Arduino looks for again and again in the loop and takes action accordingly.
Hope this helps, try this out, and do let me know the result.
NOTE
Make sure that you access the Ethernet Shield from a device within your LAN only otherwise you'll have to use port forwarding.