Arduino and ESP8266 communication over wifi
On the Uno with Ethernet shield you start an EthernetServer on port 23 or 2323. See ChatServer example.

On esp8266 you create a WiFiClient, connect it to IP address of the EthernetShield and port 23 (or 2323). Everything you print to WiFiClient will be send to EthernetServer's EthernetClient. 

You can test the server with Telnet client.