Your plants are moth orchids or phalaenopsis. They often intimidate people but are really easy to care for. The brown spike you describe is the perfectly normal result of a spent bloom. Go ahead and cut it off. These plants grow in bark, coconut husk, and spaghnum moss mixtures rather than soil. Keep it indoors where it will get lots of light but indirect sun. Let it get cooler night temps w/ open window air flow. Place it on a saucer of gravel with water to provide humidity.
Water it twice a week in the sink where the water is allowed to flow throgh the container and saturate the bark mixture, then let it drain well . Fertilize it with a fertlizer like Eleanor's VF11 at half-strength once a week. Your local garden center can help you w/ problems or repotting.
The roots grow up out of the soil and look like geen coils. A new flower spike starts out with a tip like a mitten. It can take weeks or months for a new spike, so I clip a silk orchid stem to the support stick to curb my impatience. Hope that helps:>)