Serial Communication between ESP8266 and Arduino Uno
Since you are  Arduino Uno, it has only one Serial. This Serial can be used to transmit data in USART mode to another Serial Device as well as see any Debug data in the Serial terminal.

However, both the functionality can not be used simultaneously. The Tx Rx pair of Arduino can be connected to one pair of other device i.e. either to the computer via the cable or to the Tx Rx of ESP8266.

But if you really need to plug in the Uno to the computer and send data to ESP8266 simultaneously, then you have to use SoftwareSerial library to convert any digital pin to Tx Rx pin, and connect them to ESP8266 for transmitting data.