Nestor is one of the Greek commanders and the play's official wise old man. Because he's been around the block a few times, the other characters give him a lot of props:
Let me embrace thee, good old chronicle,
That hast so long walk'd hand in hand with time:
Most reverend Nestor, I am glad to clasp thee.
I would my arms could match thee in contention,
As they contend with thee in courtesy.
I have seen the time. (4.5.202-210)
Because old Nestor has "seen the time" (he's super old), he's portrayed as a storehouse of history (a.k.a. a "chronicle") and knowledge. As such, he's a figure to be admired, respected, and cherished. But, apparently, he likes to "stroke" his beard a lot when he dispenses wisdom to his fellow Greeks, which is why Patroclus and Ulysses make fun of him (1.3.65).