If you are using an IMAP account in Outlook you will find that categories are not supported in Outlook 2013. To find out if you are using an IMAP account, click the File Tab on the Ribbon. In the Info section of the Backstage View, click the Account Setting button as shown below.
If your type of account is IMAP, then this is your issue (see below).
What is the Fix?
I fixed this problem when syncing my Gmail account in Outlook, but the fix should work for any email service. First thing I had to do was enable POP3 in Gmail. IMAP and POP3 are just different protocols, both of which allow Outlook and other email clients to access emails on an Internet server. You will need to set up Outlook to receive Gmail using POP3 as well.
Please see this page for all the necessary documentation if you are using Gmail. https://support.google.com/mail/troubleshooter/1668960?hl=en#ts=1665119,1665157,2769079
If you are not using Gmail, then your email service should provide the equivalent information.
Setting up the POP3 account in Outlook will automatically create a pst file which is also necessary if you want to use categories. A pst file is the type of data file that Outlook uses to store data. You need to make sure that your POP3 email account and the pst file are both set as the defaults in Outlook.
Go back to your Account settings and use the Set as Default button on the Email and Data Files tab for your POP3 mail and pst file respectively.
Click Close to come out of the account settings.
The second step is to enable categories in your pst file. To do this right-click on the account folder in the mail folder pane and select Data File Properties.
Now click the Upgrade to Color Categories button
Click Yes in the following message.
Categorising Email
You can right-click an email to assign a category. Microsoft has hidden the quick-click functionality for categories but you can see it if you turn the Reading Pane off.
If an email has been assigned a category it will show up in the inbox.
Categorising Calendar Items
You can categorise a calendar item when you create it. The Categorize button is found on the EVENT tab.
You can also right-click an existing calendar item to assign a category.