What kind of reputation does Romeo have in Romeo and Juliet?
I take it you are refering to what we are told about Romeo before we actually meet him in Act I scene 1. After the altercation between the Montagues and the Capulets, Montague and Lady Montague express their concern about their son with Benvolio, and Benvolio tells them how he is obviously wanting to be by himself at this time, and is shunning company:
Where, underneath the grove of sycamore
That westward rooteth from this city side,
So early walking did I see your son.
Towards him I made, but he was ware of me
And stole into the covert of the wood.
His parents share what they know about Romeo as well, and how he is clearly depressed, as reference to his "tears" and "deep sights" give witness. Something is clearly troubling him, and he has gained a bit of a reputation for being a loner and for shunning company:
And private in his chamber pens himself,
Shuts up his windows, locks fair daylight out,
And makes himself an artificial night.
Romeo therefore has gained a reputation for being dark and moody and rejecting the company of his family and friends. Whatever is troubling him has caused a significant change in his behaviour, which is why his family are so concerned.