I think it's worth pointing out that there can be some unexpected behaviour here if you don't thoroughly understand how PRIMARY KEY and UNIQUE interact.
As an example, if you want to insert a record only if the NAME field isn't currently taken, and if it is, you want a constraint exception to fire to tell you, then INSERT OR REPLACE will not throw and exception and instead will resolve the UNIQUE constraint itself by replacing the conflicting record (the existing record with the same NAME). Gaspard's demonstrates this really well in his answer above.
If you want a constraint exception to fire, you have to use an INSERT statement, and rely on a separate UPDATE command to update the record once you know the name isn't taken.