Why doesn't my replacement lighting work after my old fixture died?
It sounds like you have a fixture controlled by a conventional dimmer and are using CFL non-dimmable bulbs.

Standard dimmers were designed for incandescent bulbs. Many CFL and LED bulbs are non-dimmable. Some are dimmable, but only work with special dimmers designed for their electronic circuitry (conventional bulbs have no circuitry, only simple filaments). Some CFL or LED (not many) will work with a standard dimmer.

First you need to make sure the bulbs are dimmable, and then that you have a compatible dimmer.

In general, CFL and LED bulb packages indicate whether they are dimmable or not. If it does not say dimmable, it is probably not. Dimmers also indicate whether they are intended for CFL and/or LED bulbs. Again, if it doesn't mention CFL or LED, it is probably not compatible, unless you have bulbs that specifically say works with standard (or conventional) dimmers.

Check the fixture by using an incandescent bulb. If it works, either go with incandescent, find CFL or LED bulbs that work with standard dimmers or swap out the dimmer.

If the incandescent doesn't work, you have a wiring problem or a defective fixture.