Some varieties of orchids, such as moth orchids, bloom multiple times on the same spike. Some orchids bloom only once per stem. Whether you cut the stem after the blooms are gone depends on which type of orchid it is.
Determining the Type of Orchid
If the orchid plant has a pseudobulb---an oval growth at the base of the bloom spike---it probably blooms once per stem. This category includes the oncidium, miltonia, cattleya and zygopetalum families of orchids. If the orchid has leaves at the bottom of the plant from which the stem emerges, it probably a slipper or moth orchid, which can re-bloom from the same stem.
Cutting Non-Reblooming Orchids
If the orchid is one of the many that bloom once per stem, the stem should be pruned back. The proper time to trim the stem is after it has dried. Orchids are susceptible to viruses, so use only sterilised blades to trim plant tissue.
Cutting Reblooming Orchids
Phalaenopsis have nodes on the stem similar to asparagus. When a phalaenopsis finishes a bloom cycle, the stem should be cut halfway down, to 1/4 inch above a node. Some slipper orchids (paphiopedilums) are sequential bloomers, showing one blossom at a time for more than a year. It would be a loss to cut this type prematurely.