Wiring a GY-521 to an Arduino Uno R3?
I presume you have seen this page. If not it's probably a good place to start. All you should need to do to talk to that GY-521 chip is to hook up the I2C pins. On the Arduino UNO R3, you can see these pins, which are named SDA (for S erial DA ta) and SCL (for S erial CL ock), labelled on the back of the board, as depicted in this photo from the Arduino informational page for the board. 



You will also need to connect GND (which means Electrical Ground in electrical speak) between the GY-521 and the Arduino UNO, and you will need to 5V (which means positive five volts) to the sensor module. 

Now, I think based on this schematic:



... that the sensor module actually operates at 3.3V, so you will probably run into some voltage level problems making the two devices communicate. Luckily, this problem is easily solved with a couple of transistors as described in excruciating detail on this page (and various answers to similar questions on this site).