Blinking of Tx,Rx led's on arduino uno,constant blinking of L led

Connected an arduino uno r3 to an esp 8266 wifi module.Whenever AT commands are given in serial monitor both tx,rx leds of arduino are blinking and esp module is responding.L led is blinking on and off
Started again with a new arduino board,followed the similar steps,but only rx is blinking on arduino when commands are given in serial monitor.What does that signify?Why tx led is not blinking?And in this case the L led is constantly on...and esp is not responding.So please help with this issue....

Is there any form of program on the Arduino?

And keep in mind, the ESP8266 is a 3,3V device! No 5V for that puppy, not even on the RX line.