An external conflict is a struggle the character experiences outside of his/her own mind. While an internal conflict is a battle between character and self, an external conflict is a battle between the character and something external to the character. One example can be a character vs. character conflict in which the character battles with opposition from a second character. Other examples can be character vs. society, character vs. circumstances/fate, and character vs. nature. In Acts 1 and 3, the external conflicts Tybalt faces are both character vs. character conflicts.
In the first act, Tybalt engages in a conflict with his uncle, Lord Capulet. When Tybalt recognizes Romeo's voice at the ball, he feels extremely insulted by Romeo's presence. He assumes that Romeo has only come to mock their party. However, when Tybalt moves with his sword to avenge himself for Romeo's insult by striking him down, his uncle stops him, opposing the idea. Lord Capulet argues that Romeo is behaving like a proper gentleman and is also respected by all of Verona as a "virtuous and well-govern'd youth," as we see in his lines:
He bears him[himself] like a portly gentleman,
And, to say truth, Verona brags of him
To be a virtuous and well-govern'd youth. (I.v.69-71)
Therefore, Capulet orders Tybalt to leave Romeo alone and to not harm Romeo in Capulet's house. However, this command makes Tybalt feel even further insulted for now his own uncle is telling him to "endure" the presence of their enemy. Hence, since Tybalt and Lord Capulet disagree on how to handle Romeo's presence, we see that this is a perfect example of an external character vs. character conflict.
The conflict Tybalt faces in Act 3 stems from the same reasons as the conflict in Act 1. However, this time, the conflict is between Tybalt and Romeo. Having felt insulted by Romeo at the ball and then being insulted again by his uncle, Tybalt now wants to challenge Romeo to a duel in order to avenge himself. Since Romeo has just married Juliet and doesn't see the Capulets as his enemy, Romeo opposes the idea of a duel. Hence, since both characters oppose each other, we see that this is another perfect example of an external character vs. character conflict.