Continuous lighting uses a constant light source to light your subject, meaning that your lights will stay on during the entirety of your photo shoot. For beginners, this is often the lighting source of choice, as it offers easier anticipation of where light and shadow will appear in photos, is easier to setup (there is no triggering required by your camera), and is typically less expensive. The bulbs are cooler, meaning that your room should not get as warm as with traditional lighting. Continuous lighting is good for table top photography, and a necessity with video.
Strobe lighting uses a flash of light at the moment your camera shutter opens to illuminate your subject. This allows you to use brighter lighting than would be possible with continuous lighting, since your subject would not need to withstand the bright lights for an extended period of time. Many professionals use this type of lighting because it offers more control of the light. There is a greater ability to adjust the amount, direction, and color of light with the many accessories available (and sometimes built into) a strobe head. For example, you can use a snoot, barndoor, beauty dish, or color gel with a strobe, but doing so with continuous lighting would be either difficult or not possible at all. Strobes are ideal for action shots and photographing children. They are, however, generally more expensive.
When you make your decision, work with what you know, or are most comfortable with.