I was recently asked to outline what I thought a truly vibrant parish would look like. I thought a lot about how to answer that, and I realised that whatever answers I would come up with would be necessarily coloured by whatever latent or overt vision I had of the Church.
The Church was born on Pentecost, when the Holy Spirit descended upon the 120 gathered in the upper room. The presence of the one Holy Spirit in the hearts of all those believers united them, taking them from being a mere gang of individuals to being one Body.
A particularly significant point from that day is that the 120 included the 12 Apostles (11 originals + Matthias, elected to replace Judas). 120/12 = 10, an important number in Judaism, as it represents a minyan, the minimum number of Jewish men necessary for a prayer gathering to truly be not just a group of people praying, but an act of the people of Israel in their worship of God. So the Church, while one, simultaneously contained within it 12 minyanim, each headed by an Apostle. In other words, from the very origins of the Church, the Apostles were not merely free agents but were, by their very office, in communion with a local expression of the Church. This was expressed in later years by the idea that a bishop without a local church can't really be considered a bishop in the fullest sense, and a local church without a bishop can't really be a local church.
The Apostles went out preaching the Good News of Jesus as Messiah, teaching in the Temple area and visiting synagogues. Some Jews accepted this news, and soon new "Christian minyans" were sprouting up. Each required its own "apostle" as its head, so the Twelve began to appoint others to share the ministry with them, people today known as bishops (and who even today still retain the ancient title "successors to the Apostles"). Assisting these apostles were men appointed to act as "elders", who were vicars of the apostles and bishops and who could act in their place in certain circumstances (today known as priests), and other men appointed as deacons, given the task of organizing the practical life of the Church and of what we would call today the "apostolate of the laity".
So what was a "local church" in the earliest days? It would have been urban, with the bishop presiding the main Sunday eucharist, with the priests either concelebrating with him or being sent out to celebrate in the "sub-urbs" (i.e. surrounding smaller towns), and with the deacons taking communion to those who could not attend the Eucharist. The Eucharist would have defined the life of this local church: all missionary action was oriented towards it, and all authority flowed from it. The bishop, for example, because he presided the Eucharist (and because the priests had to work in communion with him), had the ultimate spiritual power: the power to "ex-communicate" someone, i.e. to deny them participation in the Eucharist (and by extension, in the communion of the church) for serious transgressions. The purpose, of course, was not to lord it over others, but to protect the integrity of the faith of the Apostles handed down to the churches.
Parishes began when the Church faced the situation of having large numbers of believers in the suburbs and rural areas, such that they deserved to have the permanent presence of a presider who could shepherd them, but who (at the same time) were not large enough that they could provide the necessary education and infrastructure to guarantee the excellence of their ministers and ministries. Some sort of intermediate, semi-autonomous entity was needed, somewhere between the diocese and a chapel — and the parish was born. Part of a diocese, yet also with some local autonomy, the parish became the basic structural element of the local church.
So what is a parish, then? At its basic level it is a Christian minyan, i.e. a stable grouping of the Lord's faithful, shephered by a priest acting as the "vicar" of the bishop in that area, who come together on a regular basis for the Eucharist. Beyond this, however, the autonomy the parish is meant to enjoy means that a parish also must possess a minimum vitality. Ideally, it should: (1) be motivated to always be on the lookout for new vocations to the priesthood, to ensure that the link with the original apostlic life and faith is never broken; (2) place the celebration of the liturgy at the centre of the life of the community, keeping in mind that it bears a "treasure of Tradition" in that liturgy that cannot be lightly tampered with; (3) possess an active program of Christian initiation (baptism, confirmation, Eucharist) and re-initiation (reconciliation), so that people who desire to be full participants in the work of Christ in the liturgy can receive the fullness of initiation necessary (along with the accompanying formation); (4) be financially prosperous, able to rely on the good stewardship of the members of the parish (i.e. no need to beg for money), and capable of making its proper contribution to its sister churches by means of a full contribution to the diocese + other special collections; (5) have a way to coordinate the works of the apostolate based on the gifts of talent the Holy Spirit has distributed to the members of the community; and (6) realize that the parish is itself a communion of smaller churches, in this case "domestic churches" (i.e. families/households) who need to be helped to become the centres of holiness that God intended them to be.
That's pretty much my vision: active and co-responsible vocations ministry, liturgical fidelity and renewal, catechesis designed for constant initiation into the Mysteries of God, a good sense of stewardship, a vibrant lay apostolate, and spiritual leadership to and within families. I think those need to be our priorities for our parishes — not because they sound good, but because they are part of how the Church itself is constituted on a local level. These priorities, in other words, fit the paradigm set up by the Apostles themselves in the earliest days of the Church. There is a lot more that can be said on this, of course, such as expanding on the specifics of each point, but I've given the general picture. Parishes like this can change the world.