Start a new file in Illustrator by clicking Create new… on the main menu. Alternatively, you can open an existing file to edit in a transparent background. You can drag and drop the file onto the workspace or use the File > Open menu.
To remove the default white artboard, enable the transparency grid. You can find this setting under the View tab at the top. Select Show Transparency Grid.
You will know it's working when the background appears as a white and gray grid.
Now that you have a transparent background, you can create a design. The final result will have a transparent background if saved to an appropriate file type.
If you are editing an existing AI file, the transparent background should be visible where the artboard previously was.
When your design is complete, save the changes with an appropriate file type so the transparency stays in the final product. You can save the image using the File > Export As... menu.
There are several file types that retain transparency. A few used by Illustrator are:
You may also be interested in cropping your final image.