I would define a conviction as my perception of something I should do, or not do, for which there is no direct biblical command or prohibition. It is my personal sense of what is right or wrong for me, based upon inferences I have made from the Bible. When the Bible forbids adultery, it is not a matter of conviction; it is a command, and thus it is a matter of my obedience or disobedience. (Unfortunately many professing Christians are going to this extreme today, excusing their sin because they don’t feel it is wrong, even when the Bible says it is.)
A conviction is not a preference either. We all have preferences in food, in activities, in entertainment. I may prefer chicken fried steak to split pea soup, but I am not sinning if I eat split pea soup. A conviction is a personal value, and if I violate one of my own values, it is a sin for me. If I think drinking coffee is a sin (we know that the Bible does not directly teach it as a sin), so long as I think it is a sin to drink coffee, I will sin by drinking it.
There is a great deal of misinterpretation and misapplication when it comes to the so-called “weaker brother.” The weaker brother is not just a person who thinks something is wrong, he is also the brother who is likely to do what he thinks is wrong if he sees me doing it. If someone in the church thinks its wrong to eat in a restaurant on Sunday afternoons, he may be “offended” or rather “irritated” by the fact that he sees me doing so. But he is not really a “weaker brother” unless he sees me doing this, and then he does it too, because he saw me doing it (but still feels that it was somehow wrong). The weaker brother is the one who thinks drinking wine is wrong, but he does so because he sees me doing it, and thinks that since I am mature it must be okay. The point I am trying to make it that a lot of people may be “irritated” by seeing others exercise their liberties, but they are not “offended” in the biblical sense by doing what they feel is wrong. I ought not hold my convictions over the heads of others, when I have no intention of doing what they do.