1) The universal church consists of all those who have a personal relationship with Jesus Christ. “For we were all baptized by one Spirit into one body—whether Jews or Greeks, slave or free—and we were all given the one Spirit to drink.” (1 Corinthians 12:13) This verse says that anyone who believes is part of the body of Christ and has received the Spirit of Christ as evidence. The universal church of God is all those who have received salvation through faith in Jesus Christ.
2) The local church is described in Galatians 1:1-2: “Paul, an apostle … and all the brothers with me, to the churches in Galatia.” Here we see that in the province of Galatia there were many churches—what we call local churches. A Baptist church, Lutheran church, Catholic church, etc., is not the church, as in the universal church—but rather is a local church, a local body of believers. The universal church is comprised of those who belong to Christ and who have trusted Him for salvation. These members of the universal church should seek fellowship and edification in a local church.
In summary, the church is not a building or a denomination. According to the Bible, the church is the body of Christ—all those who have placed their faith in Jesus Christ for salvation (John 3:16; 1 Corinthians 12:13). Local churches are gatherings of members of the universal church. The local church is where the members of the universal church can function out of the gifts given them by the Holy Spirit at their new birth. Some of these gifts are, encouraging, teaching, and building one another up in the knowledge and grace of the Lord Jesus Christ. (1 Corinthians 12)