Correct spelling, explanation: the base word of categories – category – comes from Latin categoria. This is a quite straightforward connection to the current form of the word, and you can clearly see that it did not change much. To make a plural form, category gets the typical suffix s, and because we generally do not use the spelling ys in English, and change this spelling to -ies instead, the plural form is accordingly spelled categories.

Definition of categories:
noun (plural of category)
1. a division of things that have something in common
We can distinguish sports into different categories; there are individual, group, summer, and winter sports and many others.
The shop offers many categories of products, from electronics to office supplies.

Incorrect spelling, explanation: the word catagories is an incorrect spelling of category because it does not fit the original spelling of the word, which is Greek katēgoria. In some cases, the vowels change due to the pronunciation or other types of assimilation into language, but in this case, the e stays in the same place in the current form: category.