What is the difference between a church, a cathedral, and a basilica?

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A church is a "house of worship," a building in which Christians gather to perform the rituals of their religion and interact with one another and hold religious functions and so on. They can be very plain, very simple.

The ruins of an ancient church.

A cathedral is a church which is also the "seat," in the bureaucratic sense more than the literal sense, of a bishop (or, in some denominations, another comparably high-ranking ecclesiastical figure). You therefore ordinarily see just one cathedral per denomination per city. Because bishops are responsible for an area —in Catholicism a diocese— a cathedral can also be thought of as the church associated with the administration of an area.

In common usage, people call really big churches cathedrals pretty often, but this is imprecise and technically mistaken.

A basilica was originally a Roman building featuring certain architectural elements that supported its use as a public, open facility for business, trading, etc. These typically —but not always— included colonnades, naves, and aisles, not unlike a modern pedestrian mall.

Floor plan of a Roman basilica; note how stalls for vendors become naves for chapels.

When some churches were built with similar features in the time of early Christianity, they were called basilicas. Later, in Catholicism, the term acquired a new meaning: the pope designates some churches basilicas for a variety of reasons, and they become important sites:
"The papal or major basilicas outrank in precedence all other churches. Other rankings put the cathedral (or co-cathedral) of a bishop ahead of all other churches in the same diocese, even if they have the title of minor basilica. If the cathedral is that of a suffragan diocese, it yields precedence to the cathedral of the metropolitan see. The cathedral of a primate is considered to rank higher than that of other metropolitan(s) in his circonscription (usually a present or historical state). Other classifications of churches include collegiate churches, which may or may not also be minor basilicas."

So basilicas as Christian buildings are mainly a Catholic phenomenon. And indeed, the world's most famous basilica is of course St. Peter's in Rome, designed in part by Michelangelo, its plaza and baldacchino by Bernini, its balcony where crowds see their pope, etc.

It's especially beautiful inside: