You can add multiple product tags for use in Products > Manage Products > View Product Tags. Click on the Add Tags button and enter different tags separated by commas. You can then assign these tags to any product.
Assigning Tags to Products
To add tags to any product, you can do that through Products > Manage Products > View Products.
Select any product checkbox on the left hand side, and in the bottom dropdown where it states Mark Checked As, you can select Add Product Tag, and a dropdown field will allow you to search for tags that you have added in your system.