Three ways to do this...
1) Remove the background
If you have a white shape in your Illustrator as background, then delete it. I'm sure that's not your issue. By default, the background of Adobe Illustrator is transparent even though the artboard is white.
2) Check the box "transparency"
When you do a "save for web", don't forget to check the box "transparency" when saving a gif or a png file.
3) Remove the color with the color picker
If you still have a background even if the box "transparency" is checked, then use the color picker tool on the top left of your "save for web" window, and click in the white background. Then, go on the right side where the color chart is, and click the first small icon; it will change the color you previously selected to transparent.
You can also do this by simply clicking on the white swatch on the right side color chart, and then clicking the small icon I mentioned above. You can do this for as many color as you want.
Note that this technique will remove ALL the white or the selected color on your gif!
If you don't want to see the A4 at all and only the graphics, you can uncheck the "clip to artboard."