Two helpful, but not biblical, labels
The Bible does not use the exact phrases “visible church” and “invisible church”. Christians have used them to make sense of two truths Scripture holds together. Jesus has a real people in the world, gathered in recognisable communities, and he also knows his people at a depth no human can see.
In the New Testament, the church is not merely a building or a weekly event. It is Christ's people. Paul could call the believers at Corinth Christ's body, even though that particular congregation had serious divisions and failures (1 Corinthians 12:27).
Now ye are the body of Christ, and members in particular.
What people mean by the visible church
The visible church is what can be seen: local congregations, gathered worship, baptism, teaching, prayer, fellowship, and care for one another. After Peter preached at Pentecost, those who received his message were baptised and joined a community devoted to the apostles' teaching and shared life (Acts 2:41-42).
Because churches are made up of ordinary people, outward membership and inward faith do not always match. Someone may attend, speak Christian words, or hold a role, while their heart is far from God. Paul expected the Corinthian church to address public, unrepentant sin, which shows that the gathered church has real responsibility, even though it cannot read hearts (1 Corinthians 5:12-13).
Then they that gladly received his word were baptized: and the same day there were added unto them about three thousand souls. And they continued stedfastly in the apostles' doctrine and fellowship, and in breaking of bread, and in prayers.
What people mean by the invisible church
The invisible church does not mean a secret club or Christians who never meet together. It means the whole family of everyone who truly belongs to Christ, including believers in many countries, traditions, and centuries. Their faith may be visible in its fruit, but only God sees every heart without error.
Paul gave Timothy confidence in this when some people were departing from the truth: God's firm foundation stands, and the Lord knows those who are his (2 Timothy 2:19). Jesus also speaks of his sheep as those he knows and gathers, including people beyond one immediate group (John 10:14-16).
Nevertheless the foundation of God standeth sure, having this seal, The Lord knoweth them that are his. And, Let every one that nameth the name of Christ depart from iniquity.
Why the distinction matters
This distinction should make us humble. We should not casually decide who is or is not a genuine Christian, because God alone has complete knowledge. At the same time, it does not make public church life optional. Believers are urged to keep meeting together so they can encourage one another in faith (Hebrews 10:24-25).
So it is good to belong to a faithful local church, to worship and serve alongside real people, and to test teaching by Scripture, as the Bereans did (Acts 17:11). No congregation is perfect. Yet Christ is faithfully building his people, and he will not lose those who truly belong to him.