"The country respects the president."
Translation:El país respeta al presidente.
The "a" is there to indicate that the president is the object of the sentence. Just like you would say "Juan respeta a María." That shows that Juan is the one doing the action and María is the receiver. "María respeta a Juan" would mean that Marïa respects Juan. Juan in this case would be the object of Mariá's respect. The personal "a" is used a lot to support the person who is the object of the sentence, to distinguish between the subject and the object. It is useful in questions, too. "¿Respeta María a Juan?" Both names come after the verb, but the "a" tells you that María is the subject and Juan is the object.
The president is the direct object. If you said, "I sent a card to the president," then the president would be the indirect object. "Envié una carta al presidente." In the sentence, "You saw the president" it would be "Tú viste al presidente" with the personal "a" showing that the president in this case is the direct object and not the subject.