1. Change to person view rather than your default Organisation view by clicking on the Refine Search icons as shown below: 

2. The icon on the left with a single person is for individual view and the icon on the right is for Organisation view - here you can still search for a company but the people contacts are shown and here you can select the person and tag the contact appropriately.

3. Search for the contact record you wish to apply the tag for

4. Open up the contact record and click on the green tick tag icon as shown below:

5. An Edit tags window pops up.  Apply your tag(s) by placing a tick in each tag name box that you wish to apply.

6. Click on Update to save your changes.