Your basic church takes its cue from the Greek origins of the word assembly and also the phrase "belonging to the Lord." If a building is consecrated to assemble the faithful for worship (those also known collectively as the church), and if the building is therefore a "house of God," then it's a church.
A cathedral, by contrast, is the particular church in which the bishop presides over worship and, by extension, over the diocese at large. Historically, cathedrals were grand works of art that took centuries to build: Those who began the construction rarely saw its completion. The cathedrals of Europe were vibrant centers of urban life and learning. Cathedrals tended to be larger than the average church, although the trend toward mega-churches in parts of the United States today have made some local cathedrals seem diminutive by comparison.
Basilicas were originally official buildings of the Roman Empire; the Greek word means "king's hall." When Christians acquired these buildings they were appropriated for Christ the King. These historical structures include four major basilicas of Rome: St. John Lateran, St. Peter's, St. Paul's Outside the Walls, and St. Mary Major.
One might imagine there could be no such thing as a modern basilica by definition. But minor basilicas continue to be named according to a church's historical significance for a particular region. At present more than 1,500 basilicas encircle the globe, with four countries—Italy, France, Poland, and Spain—garnering over 100 each. Five cities enjoy more than ten basilicas, including Rome, Buenos Aires, and Krakow. Jerusalem and Paris each have five. The U.S. has 65 basilicas, including the Cathedral of St. Augustine, Florida, where the first American Catholic parish was founded; Mission Dolores in San Francisco; Sacred Heart Basilica on the campus of the University of Notre Dame; and the Baltimore cathedral.
If churches are intended to gather all the faithful to worship, chapels (sometimes called oratories) serve more specific populations. Folks stuck in airports appreciate the terminal chapel; prisons, hospitals, schools, convents, and religious houses also have chapels for their communities. Each bishop has the right to an oratory in his residence. In addition, some churches have a smaller chapel attached for daily use.
Exodus 3:4-5; Isaiah 56:6-8; Psalms 24, 42, 84, 95, 100, 122, 133; Mark 11:15-17
Built of Living Stones: Art, Architecture, and Worship, from the United States Conference of Catholic Bishops
Theology in Stone: Church Architecture from Byzantium to Berkeley by Richard Kieckhefer (Oxford University Press, 2008)
How to Read a Church: A Guide to Symbols and Images in Churches and Cathedrals by Richard Taylor (HiddenSpring, 2005)