Product attributes need to be defined first before they can be added to a product. You can create definitions under the Catalog > Attributes definitions menu.
You can classify the new attribute definition to one of the groups you created earlier under Catalog > Attributes groups.
It's important to pick the correct Attribute type (Boolean, Decimal, etc.) as it determines how it will be used in product comparison and for filtering products. For example, if you sell televisions, you may have an attribute definition called "HD Ready" and it would take the Boolean attribute type because the possible values are either Yes or No. Whereas, you can have another definition for "Weight" and it would take the Decimal attribute type because the weight can be any number.
You can also set the attribute definition to be Published if it should be shown to the customer, Comparable if it should be used in product comparison or Filterable if it should be used to filter against the product list.