Yes, one could describe many "purposes" of the church, universal as well as local.
1. The Lord taught us to pray "... thy kingdom come, thy will be done on earth as it is in heaven..." The church is to demonstrate the kingdom of God on earth. "Seek first the kingdom of God and His righteousness, and all these things will be done unto you." The church should be the place where all people can see the "righteousness of God" in action - within and outwardly. Israel, as a chosen nation was to show the other nations, how a Godly nation would operate, based on God-given principles and justice. The church should be this light on the mountain top. (NOT the power on the top!) It should be the salt - the conscience - of our various countries, through high moral and ethica living beyond any national legal requirements. The church should be in a way, the equivalent of "ten righteous men" living in Sodom and Gomorrah, for whose sake God would spare those cities.
2. Jesus told the disciples "A new commandment I give unto you: love one another." and "They (the world) will know you are my disciples if you love one another." In the westernised world, the individual is the focus; most western societies have lost the true understanding of community - and those few places that try to practise it, are heaped with accusations (many justified) - usually because individual "rights" are not given first place. You cannot live an individualistic (self-centred) life in love. There must be a group, a community, of some kind in which to demonstrate love. And not only love - most, if not all of the fruit of the Spirit cannot be demonstrated in isolation from others. The church provides the community in which to develop and demonstrate love. Again, not only inwardly but towards outsiders, including "our enemies." It is much easier to love your enemies if you do it with others.
3. Paul wrote to the Ephesians (ch 4:11-14) " So Christ himself gave the apostles, the prophets, the evangelists, the pastors and teachers, to equip his people for works of service, so that the body of Christ may be built up until we all reach unity in the faith and in the knowledge of the Son of God and become mature, attaining to the whole measure of the fullness of Christ." When I first examined this verse I thought that every one of us would attain the whole measure of the fullness of Christ (and remember "in Him rests the fullness of the Godhead bodily"). Now I think it means that we together - as the church - will attain this fullness...and to the extent that we become united together and with Him. The church may be seen as multi-generational, and is being perfected through time; but embedded in this development process, this maturing, is I believe also the principle that the church should be doing all in its Spirit-given power to nurture all believers and to provide the context in which every member is able and encouraged to develop her or his God-given gifts, whatever they are.
4. Others have mentioned other responsibilities that the church has- such as evangelising, of being the salt and the light in the world around, and I do not disagree with any of these. The above are my additional thoughts on the subject.
My own experience is that neither the universal nor the local church is really living up to its purpose(s). Therefore I pray regularly and earnestly that the Spirit will move all believers (including myself) to confess our individual and collective failures to the Lord (think of how Nehemiah and Daniel identified with, and confessed the sins of their people, both the current and the previous generations); because before we can ask for widespread revival, we must seek forgiveness and internal healing. We do not display unity, nor love, nor community, nor the kingdom of God in many of our local churches or various "denominations." Shame on us! (But thanks be to God, who loves us and never gives up on us, and to Jesus who gave his life for his "bride," the church, and strives to present her spotless to the Father.)
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