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I have a '97 Ford Taurus that has been running fine up until a few weeks ago, when it seemed I needed to jump the battery almost every time I started the car.  One day, it just wouldn't jump, so I had a friend look at it and he told me that the starter had gone bad.
I bought a Haynes repair manual and had him look at it, and then bought a brand new starter, which my friend installed for me.  After installation, which he *** me he did correctly, there is still something wrong.  When the battery is connected, the starter keeps running and won't shut off, and after ignition, it does the same thing.  When the car is running and the battery is disconnected, the car keeps running off the alternator, yet the solenoid won't turn off.  (If I'm getting my terms mixed up, it's only because I'm not great with cars.. this is what he told me is happening).
Anyway, we called a Ford dealership, and one of the mechanics told us to look for a relay switch on the starter lines, but there aren't any, and now we're stumped.  Anyone know what might be causing this problem?  

Sounds like a starter relay sticking. When it continues to crank bang on the the starter relay and see what happens. Bad relay or solenoid will do that.  

Was this problem with the starter, continously running, happening before you changed the starter?
If not you may have the + cable touching the  small terminal on the starter solenoid. Check your wires.