Also, before connecting up to your Raspberry Pi or whatever, connect it up to a PC to verify it.
Communicating with the ESP8266 module in AT mode requires carriage return and line feed to be appended to each command.
A short while into the sequence then the module detects that is a 26MHz crystal and adjusts the baud rate to 115200. You should then be getting the rest of the message in plain text if the terminal is set to 115200. For a raw module with AT firmware this will have a reference to AT firmware and a ready message and should then accept AT commands.
Things that might be going wrong include:
1) Bad power as suggested.
2) Pins not configured to boot into normal run mode. GPIO15 should be pulled down to 0V via say a 1K resistor. EN (Enable should be connected to 3.3V. GPIO0 and GPIO2 should be at logic level 3.3V. Normally they will float high with internal pull ups but you can externally pull them up.
3) TXD0 and RXD0 connected wrong way round.
Many people use a USB serial adaptor to connect a PC to the TXD and RXD. These work well but you should not use the 3.3V offered by these modules as it can't supply enough current. I actually cut the 3.3V connection and put a 1117-3.3 regulator (fed from the 5V on to feed the pin instead. This gives a convenient way to power the esp.