ESP8266 arduino wireless.

Short answer yes with some notes:
To use the ESP8266 you'll need to parse the strings you get from the device in order to get the data. The ESP will output different messages about status and things like that that you'll need to handle. All data receive from the ESP will be on a line that starts with +IPD followed by message lenght then data terminated with carriage return line feed.

With the ESP8266 you'll need an Access point or one of the them to be an access point. One of the module should be a server and the other a client. Then data can be sent from one to the other. This is a fairly new device, I think it supports direct Wifi but I have not use that.

If you want a cleaner serial communication between two Arduinos and they are near one to the other, it may be easier to use a Bluetooth to UART module. Or if they are farther away an NRF24 wireless module is nice (range depends on environment), or maybe an Xbee (xbees tend to be more expensive). The bluetooth modules and the NRF25 are reasonably cheap. The NRF normally a little cheaper than the bluetooth.

Do not get me wrong I am in love with the ESP8266 so I will encourage you to give it a try, it should work. You'll need some level shifting (a few resistor or else) to change 5v to 3.3v signals at least on the arduino TX signal (ESP's RX).
This is just my opinion, I am no expert. Wait for others to reply before deciding.

Anej,

          As has been said the ESP8266 is a relatively new device and there is not a lot of information on its use yet.  For your project I would use the CISECO wireless modules. CISECO do a wireless shield that fits straight onto an UNO, MEGA or one of the clones.  These modules cost about £10 each in the UK.  They have very good range and are very easy to use, no level shifting is required because it is all built into the shield.  There is very good supporting documentation and code examples on the CISECO web site. 

I would tend to disagree - there is now a lot of information - there are several projects including the original ESPRESSIF AT code which now works as of the latest updates - the FRANKENSTEIN code and a Lua interpreter (links here in various places) - sure they are works in progress but for example on the original Espressif code I now have a mobile App controlling several outputs and monitoring inputs and I can get it to work utterly reliably on both the AT command set and (if I don't get too ambitious) on the Lua interpreter - and as far as the latter is concerned the bugs are dying away by the day. There is even apparently an RTOS implementation coming along! £10 is too much unless you absolutely have to have WIFI - you can get an Ethernet card for half of that...  at £2.20 or so the ESP-01 is an unbelievable bargain and worth persevering with.  Links here if you're interested.  There's a fellow in here who has produced with Eagle an interface board that handles the level shifting and I'm doing my own. Armed with the EAGLE files anyone can get bare boards these days cheaply so I would imagine before long there will be a host of interfaces for the ESP-01 - worth sticking with if you're just experimenting right now.

Hello

I would use Aurel modules which have a very good quality-to-price ratio with the Arduino VirtualWire library

Thanks amgalbu

I was about to ask a similar question on here and was looking at either the XBee, the Laird PRM133, or the Aurel modules. Will give the Aurel modules a try now.

Not sure what you want to use these for but they are awfully simple and are easily blocked by any old radio remote control. Suggest RM22 or similar. NRF24L01 don’t have much range – but they are cheap.

There is now a fork of the Arduino IDE for the ESP8266. This allows you to use existing arduino code and libraries (Not all) to compile your own firmware for them.

With an I/O expander, or using i2c you could even do without the arduino board completely.

https://github.com/esp8266/Arduino