It's possible to delete tags from individual contacts in the following ways:

1. The quickest way to delete a single tag is to click on the "x" of the tag you'd like to remove.

2. Another other way to do it is to click on the Name to open up the contact details.....

then click on the "x" in each of the tags.

3. The last way to do it is to click on the Edit icon.....

then select the tags you'd like to remove.