Going through the book of Acts in a sermon series has been a refreshing reminder of the life of the church. Inasmuch as everyone talks about wanting to "do church like the first century church", we usually don't hit the mark. I can't help but think that we are seriously overcomplicating it.
It seems like there are at least three things we can say, keeping things pretty simple.
First, the church is founded on Jesus Christ. For any Christian, that ought to be the obvious one. You don't have the church without Jesus.
Second, the church is powered by the Holy Spirit. This message is consistent throughout the life of Jesus and of course carries through in the book of Acts. Jesus talks about the coming Holy Spirit to the disciples and then prays for them in John 17 that they would be sent out into the world. The order is important. In Acts, Jesus tells them to wait until they have the Holy Spirit. When the Holy Spirit finally comes on the day of the first fruits of the harvest, the disciples receive the promised power and they go out and spread the message of the Gospel to massive response.
Third, the church is recognized by it's community. It seems to me that everything else that happens in the book of acts, and certainly the first description of the church in Acts 2, has this as it's center. The church restores bonds between people that were previously broken.
If I were to add a fourth, it would be that the church is sustained by it's prayers. I don't like the word "sustained", necessarily, because I think we are sustained by Jesus/the Holy Spirit but it's because of that sustenance that we ultimately pray. If the church recognized that we were totally dependent on Jesus work and the Spirit's power, we'd do a lot more praying. Prayer is one of the ways we activate the Spirit's work (activate here being used loosely; I don't intend to treat it like an on/off switch or something we can ultimately control.)
When we see what the church is doing, then, what we notice is that they are taking great care to remain on their foundation–that is, they are studying the Scripture that tells them about Jesus–they are continually seeking the Spirit–prayer–and they are doing all of this in community with one another. That community, in some cases, simply means that they have decided that doing life together is better than doing it apart. They hang out, they love one another, they want to be with one another, they weep with one another when necessary and laugh with one another when they can. They share amongst themselves. There isn't need in this community, because there would never be need in a family. And we are the family of God.
This is the fundamental essence of the church of Jesus Christ, who is bound together by Him and through him, and grows together into a family by the power of the Holy Spirit.
It's easy to miss the simplicity of what the church is called to be.