"One person believes it’s all right to eat anything. But another believer with a sensitive conscience will eat only vegetables. Those who feel free to eat anything must not look down on those who don’t. And those who don’t eat certain foods must not condemn those who do, for God has accepted them. Who are you to condemn someone else’s servants? They are responsible to the Lord, so let him judge whether they are right or wrong." - Romans 14:2-4 (NLT)
The above phrase "sensitive conscience" is translated by the NIV version of the Bible as "weak in faith". This is almost certainly an extremely poor translation, since it is the only translation which calls vegetarians "weak in faith".
Another indication that it is a mistranslation is the fact that Paul considers faith essential for salvation. Also, he just esteemed vegetarian Christians as fellow believers. So to call them "weak in faith" would be to disregard his previous statements that these vegetarians are saved through faith.
With that being said, if the NIV translation is the only correct translation, it is also possible that Paul was addressing an attitude of arrogance among certain Christian vegetarians. If these vegetarians were becoming arrogant about their diet, then their "weak faith" was leading them to condemn other Christians.
Even more likely, Paul may have been addressing people who wouldn't eat meat sacrificed to idols. People are vegetarians for different reasons, and Paul may have been saying that avoiding meat sacrificed to idols was not essential (but also not worthy of judgment).
It's important to note that the Bible is filled with vegans, and not once does God condemn them (Link). In conclusion, it's funny how some Christians use this Bible verse to arrogantly condemn Christian vegetarians, when Paul's entire point was exactly the opposite. "Those who feel free to eat anything must not look down on those who don’t."