I am trying to replace an existing stairway light switch with a dimmer switch.

The light switch is at the top of my basement steps. I have no switch at the bottom of the steps, so I had assumed that the switch was single pole. When I removed the switch plate, I saw an extra wire going into the back of the switch (in addition to the hot, neutral and ground wires). I surmised that this switch was wired to be 3-way, but the other switch had never been installed. I bought a 3-way dimmer switch and installed it.

After installation, the dimmer switch turns the light on when in the "off" position and does not dim the light at all. What could be causing this?

Lets first do more looking for that missing switch. It would be nice to find it at the bottom of the strairs but we already know you weren't that lucky. Open all switch boxes in the basement to help locate another 3way, for all we know this could even been a 4 way at this point. Did the instructions asy this dimmer could be at either end of the traveler?

The instructions don't seem to indicate any particular location for the switch, so I would assume that it should not matter where the switch is added. Not sure this helps, but this is essentially the switch I have: http://www.buyhardwaresupplies.com/?t=5&m=g1&itemNumber=32217.

As far as other switches in the basement, it is difficult to test that since the basement is now finished and has its own set of switches for cans and pendants throughout. I looked through some old photos and can tell that there never was a switch at the base of the steps or anywhere else for that matter (it was all concrete slab, so no place for a switch other than the steps). The electrician who did the basement told me that the wires were not run to the bottom, so he was not able to add a switch at the base of the steps to control the stairway light. I figured I had other alternatives, so I didn't ask him to look into it further. (I now wish I had though).

How many cables in the switch box? There should be either two 2-wire (black, white, ground) cables or one 2-wire cable. A single pole switch would have only two terminals or wires for connection. A 3-way switch would have 3 connections. If there is a 3-wire cable it is possible that it was the stairs were wired for 3-way switches and one of the boxes got covered up by the dry wall installer. It would be unusual not to have a switch at both the top and bottom of stairs.

Describe the cables.

There were 3 wires and a ground wire. The 3 wires were all black with no noticeable markings on them to indicate hot, neutral, etc. Two of the wires were attached to the side of the switch and were coiled around screws. I assume that these are my hot and neutral wires.

The other wire was going directly into the back of the switch at the top. There is also a place to insert another wire on the back bottom of the switch. I had assumed that this wire was going to the missing switch that should be at the bottom of the stairs.

So you have only one cable with 3 wires and a ground, correct?

(A wire is a wire [insulated] solid or stranded.
A cable is two or more wires in an outer covering.)

Just a quick clarification before the real fix is found. "side of the switch and were coiled around screws. I assume that these are my hot and neutral wires." No, no Matt, no neutrals go to a switch, a switch gets a hot to one side and the other side becomes an extension of that same hot once its flipped. That black leaves the switch box and travels to the fixture and is attached, the neutral is the white which is the other wire attached to the fixture. Just want to correct your thinking. So the elctrician says "electrician who did the basement told me that the wires were not run to the bottom, so he was not able to add a switch at the base of the steps " did he say where he thought the extra wire was going?

More information is required , you say that there are only three black wires in the box ? Dose the box have conduit to it, or am I miss reading again.

Neutral wires don't normally go to switches, white is sometimes used for a Hot. I would try to make it work with just the wires or a switch, using only 2 wires, then install the dimmer.
If you miswire a dimmer and switch a hot to a Neutral you will destroy the dimmer.
Get it to work without the dimmer first.

Sorry about the confusion - I clearly don't know much about wiring.

I spoke to someone last night and I think got a clearer idea of what I need to do. I have the 2 black wires going to the side of the switch, the one black wire going into the back and then the ground. The person I spoke to thinks that the extra wire going into the back of the switch is either giving power to something or getting it from something. He suggested that I figure out which two wires were hot and wire those and hot wire from the dimmer together. Then wire the remaining black wire to the other wire on the dimmer. I just need to get a functioning voltage detector and will give that a shot.

If you provide a picture, it may help.
Do test by making and breaking the connection witout the dimmer, without the switch if you can safely do this.