Chilling is needed to induce flower bud initiation within the daffodil bulb. When planted in fall, the bulb sends out roots to obtain moisture, and the chilly soil causes the internal bulb tissues to form an immature flower bud. If inadequate cool temperatures occur, the bud doesn't form and the bulb only sends up leaves in spring.
As a general rule, daffodil bulbs need exposure to temperatures between 35 and 45 degrees Fahrenheit for eight to 12 weeks. This occurs either outdoors in the soil over winter or if placed in a refrigerator to satisfy the chilling requirement, known as vernalization.
The American Daffodil Society mentions some tazetta-type daffodils, such as the paperwhite, need diminished or no chilling. They do need a dormancy induced by dry soil before flowering after being planted. In the Deep South where winters are too warm for vernalization, these daffodil types or "low-chill varieties" work best.