You’ll first need to ensure the image you’re working with is a vector. If it’s a picture, you can turn it into a vector by going to the top menu and choosing Window > File Trace.
Next, we need to enable transparency in order to delete the background, otherwise it defaults to a plain white background. To do this, go to the top menu and navigate to View > Show Transparency Grid. The Shift + Ctrl/Command + D keyboard shortcut will also enable transparency.
Select the image and go to Object > Expand from the top menu to show all of the points in your vector image. Select only the points that pertain to the background you want to remove.
Once you’ve selected the background, press Delete on your keyboard. This will remove anything that was selected and replace it with transparency.
Now that your background is removed, you can save and export the image or continue your work in Illustrator.