There are two opinions as to who the weak refers to. Some say that the weak are Jewish Christians who want to keep some Jewish traditions. Others explain that the weak are ex-idolaters who were converted to Christianity and had some problems with things that reminded them of their pagan background. In the Roman context, I think it fits best that the weak are Christians with a pagan background.
1 As for the one who is weak in faith, welcome him, but not to quarrel over opinions.