Go headless (no monitor, you can RDP in) , and go with an low-power CPU, like an Atom or Fusion, as recommended by Lucas. Its the hardware and how heavily it's utilized that determines electricity costs.
There's a fairly big niche market these days in low-power servers, so you should look into that (use the Google). A lot of them are meant to run Linux, but some are compatible with Windows, or can have Windows installed as an option. If you get a plug server or a similar small machine designed to minimize power use, you can run it for literally pennies a day.
I've got a couple around the house somewhere and they cost less to run than my desktop... which really isn't all that much, when you do the math. Say, 850W (which would be the maximum draw of my desktop PSU) *$0.13*24 == $20.40 a day, if my computer was running at full power draw 24/7. It actually draws around 5% on average, so it costs me about $350 a year to keep my desktop on full time. A home server, with a similar expected utilization would cost about the same, unless you have wildly different prices for electricity. One of my actual enterprise-grade servers (A Dell NX3100) only runs me about $500 a year in electricity, or less than $10 a week, so you really shouldn't expect a server by itself to blow out your electric bill.