*There are some dendrobiums  called "nobile types." These produce flowers directly from the plant pseudobulbs. When the flowers fade simply remove them, or allow them to fall off. Do NOT cut off the old drier pseudobulbs as they are part of the plant and may rebloom the following season.

*There are an increasing number of  tropical paphiopedilums  that produce two to five flowers sequentially and can continue to bloom from the tip of the flowering scape. As flowers drop off, the nodes form a 'zig-zag' pattern. These flowering stalks should not be removed until they have turned brown.   

The rule below probably applies to most orchids.

"When all the blooms on your orchid plant  have faded, cut off  flowering stem about an inch from the base, or as close to the base you can easily get to."

Use a sharp knife and cut the stem at an angle—that's it! Once you've made the cut, discard the flowering stem. The remaining stem "stub" will dry up and fade on its own. Do not attempt to fully remove it, as you'll probably damage the plant. This applies to most of the commonly available orchids including:

Phalaenopsis (moth orchids)
Cattleyas (and related genera: e.g. laelis,slc, blc)
Dendrobium (applies to phalaenopsis type only)*
Oncidiums (and related intergenerics, e.g. Odontocidiums, Miltassias, Brassias, Colmanaras etc)
Paphiopedilums (slipper orchids)*
Cymbidiums

An old flower brown stem can be seen to the right on this 
phalaenopsis pictured above. As you can see, a clean cut was made  about an inch from the base of the plant. Two new flower spikes can be seen  to the left.  See how to grow phalaenopsis orchids for more information.