In the left side panel of that page----click on the "name" of the category. When it opens----click on the name of the contact you want to remove from it, and then click "remove from category"---don't click on "delete" or it will remove it from your address book as well.
To put that name into another category.
Open your address book----click that name---click the "add to category" tab , and in the drop menu, click on the category name you want it in.
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Click on "contacts" and your address book opens.
In the left side panel of that page----click on the "name" of the category. When it opens----click on the name of the contact you want to remove from it, and then click "remove from category"---don't click on "delete" or it will remove it from your address book as well.
To put that name into another category.
Open your address book----click that name---click the "add to category" tab , and in the drop menu, click on the category name you want it in.