The 'Attn.' or Customer Contact field allow you to specify a contact within your customers business.
This is usefull if you customer has many employees and you need to directly specify on the invoice who within your customers business this invoice is for.
ie. Within the customer 'Springfield Power Plant' you may want to specify Mr Burns (or Smithers) as the customer contact as they are the person who gets the invoice.
So an Invoice will look like
Customer: Springfield Power Plant
Attn.: Mr Burns