Correct spelling, explanation: the word is an adverb, which comes from an adjective local. As the rule of forming adverbs in English clearly states that we need to add an -ly to an adjective, in the case of this particular word we need to double the letter -l to form the correct form locally. When it comes to the history of the word, it has Latin roots, in the form locus, which meant a place.

Definition of locally:
1. adverb – occurring close, in the nearest environment,
The region is famous for beautiful china, which has been produced locally for hundreds of years.

The word family of locally:
In the word family of the word locally, we will find: local (an adjective), location (a noun), localization (a noun), localize (a verb), and many other words.

Incorrect spelling, explanation: it is quite a common mistake to add only the letter -y to the adjective local when creating an adverb, but it is wrong as it is against the most basic rule of forming adverbs. According to the rule, one has to add an -ly ending to an adjective when he or she needs an adverb. Therefore, in the case of the word local you need to double the letter -l, which gives the correct form locally and makes localy always incorrect.