Simply defined for our purposes, edification is the building up of the body of Christ. It is clear that edification is a work that Christ demands from His church as a whole.
Now concerning food offered to idols: we know that “all of us possess knowledge.” This “knowledge” puffs up, but love builds up. (1 Corinthians 8:1) (ESV)
And he gave the apostles, the prophets, the evangelists, the pastors and teachers, to equip the saints for the work of ministry, for building up the body of Christ, until we all attain to the unity of the faith and of the knowledge of the Son of God, to mature manhood, to the measure of the stature of the fullness of Christ, so that we may no longer be children, tossed to and fro by the waves and carried about by every wind of doctrine, by human cunning, by craftiness in deceitful schemes. Rather, speaking the truth in love, we are to grow up in every way into him who is the head, into Christ, from whom the whole body, joined and held together by every joint with which it is equipped, when each part is working properly, makes the body grow so that it builds itself up in love. (Ephesians 4:11-16) (ESV)
In the first case, Paul is talking to individual Christians about eating meat which makes it clear that this is outside of the context of the local church. In the second case it is also clearly talking about the work of the universal church (the body of Christ) through individual Christians. It is each Christian’s responsibility to build up his brethren by how he lives his life and in his teaching.
What then, brothers? When you come together, each one has a hymn, a lesson, a revelation, a tongue, or an interpretation. Let all things be done for building up. (1 Corinthians 14:26) (ESV)
Now we see the context of the local church. Paul says that all things done in the assembly must be done for building up the church — for helping the members of the local congregation grow in Christ. We also see here some of the ways that the local church is to edify. Our singing is meant to build each other up. Our sermons and classes based on the scriptures are meant to build each other up.
In the verse from Ephesians 4 quoted above, we see the gifts that Christ gave for the building up of His body. Most of these gifts apply to the church as a whole, but one (pastor) applies specifically to the local church. Apostles and prophets revealed the word of God as they were inspired by the Holy Spirit. We have access to their work now through the Bible. Evangelists often work in the context of a local church and they do a great work in building up the body of Christ in the location where they are working. Pastors are directly tied to local congregations in their work. They have a wonderful work of guiding the local group in their learning and work. Teachers may work in a local congregation or work house to house just as the evangelists.
The word of God is the tool that all of these people use. Apostles and prophets revealed the word of Christ, and the others continue to teach it to their brethren to build them up.
The goal is that every member of the body of Christ will be able to stand on his own and not be deceived by lies and human wisdom.