It's possible to clean parts of stove burners without removing them, but some parts of the burner cannot be accessed unless they're removed. The area under the burners is also difficult to reach when the burners are in place.
Fortunately, most electric burners are easy to remove. Simply pulling the burner to the side is generally enough to remove it. Burners can be cleaned using soap and water, and especially difficult spots can often be removed with a mixture of baking soda and paste. Commercial cleaners designed to use on electric burners work as well. It's important to make sure the burner is completely dry before reattaching it to avoid damaging electrical components within the range.