First: check whether you have corretly written the image to the SD card.
Then, if you have access, can buy or build a serial cable for your device (various instructions elsewhere) you could connect to serial e.g. with PuTTy and see what's going on.
Otherwise hard to find out.
Thanks for you answer.
1/ How can i check if my image has been correctly written on the SD card ?
2/ Ok, i'm using a laptop so i cannot see what's wrong there. But i have a TV screen with a hdmi port. With a Serial cable + a VGA/HDMI adapter, will it works ?
First of all, the current firmware is 1.0.4 so you should update it. You can follow this guide to do that: Intel® Galileo – Update Firmware
Once you have updated the firmware you can try using the SD Card Linux image. It can be downloaded from here: Intel® Galileo Boards and Kits — Software Downloads
To test if the board is booting from the SD Card Linux image you should have access to the internal Linux terminal. For this, you have to use a serial cable like the one described here: Intel® Galileo Boards and Kits — Console Serial Cables. If you don't have it you can build one by yourself (or buy one). Here is a guide about how to build one: Intel Galileo - Introduction - Malinov Family Web Presence. The cable and the adapter that you are pointing won't work for this purpose.
Another thing you can try in order to know if the board booted fine from the SD Card image is to check if the sketch is permanently stored. As you have noticed, the sketch uploaded is not stored permanently and after a reset you have to upload the sketch again when you are not using the SD Card. Using the SD card allows the board to store the sketch on the SD Card and run it after a reset. Here are some troubleshooting steps if you have issues with the SD Card: Intel® Galileo Boards and Kits — SD Card Troubleshooting
I hope I have answered your questions. Please update the firmware version and then try using the SD Card again.