Romans 14 addresses this question in a practical way. It largely assumes that the reader already understands what is meant by weak faith, but the chapter still gives insight into what Paul meant by "weak faith".
(All quotes below are NASB.)
1 Now accept the one who is weak in faith, but not for the purpose of passing judgment on his opinions. 2 One person has faith that he may eat all things, but he who is weak eats vegetables only. 3 The one who eats is not to regard with contempt the one who does not eat, and the one who does not eat is not to judge the one who eats, for God has accepted him. 4 Who are you to judge the servant of another? To his own master he stands or falls; and he will stand, for the Lord is able to make him stand.
Paul mentions the specific case of dietary laws, but a generalization of his view presented here is that the person with weaker faith has a bigger tendency to be legalistic and the person with stronger faith is comfortable with a larger variety of activities. The bottom line though is that neither brother should allow difference in faith to affect their view of the other.
Both in verse 3 above and verse 10 below Paul points out that the person with weaker faith tends to judge others (for their "improper" behavior) while in contrast the person with stronger faith tends to view others with contempt (for their lack of understanding of their freedom).
If you have a tendency to judge others or if you have a tendency to view others with contempt, you need to consider who the ultimate judge is.
10 But you, why do you judge your brother? Or you again, why do you regard your brother with contempt? For we will all stand before the judgment seat of God.
11 For it is written,
“AS I LIVE, SAYS THE LORD, EVERY KNEE SHALL BOW TO ME,
AND EVERY TONGUE SHALL GIVE PRAISE TO GOD.”
12 So then each one of us will give an account of himself to God. (Emphasis added.)
So what's the practical application/result of this understanding of "weak faith"?
13 Therefore let us not judge one another anymore, but rather determine this—not to put an obstacle or a stumbling block in a brother’s way.
22 The faith which you have, have as your own conviction before God. Happy is he who does not condemn himself in what he approves. 23 But he who doubts is condemned if he eats, because his eating is not from faith; and whatever is not from faith is sin.
- Live your life according to the convictions God has given you.
- Do not judge the convictions of a brother in Christ.
- Always be willing to yield your behavior to the stronger convictions of a brother so that you will not cause him to stumble.