If I build a magnet by wrapping a copper wire around a nail, and
attaching the wires to a battery, does the resulting electric magnet
have a pole like a natural magnet? And would it reverse if I reverse
the battery/wire connection?

Hello curiouscalifornian-ga

Thank-you for your question.

There is an excellent tutorial on the HowStuffWorks.com website
detailing how electromagnets work.  This can be found here:
http://science.howstuffworks.com/electromagnet.htm

"For example, if you wrap your wire around a nail 10 times, connect
the wire to the battery and bring one end of the nail near the
compass, you will find that it has a much larger effect on the
compass. In fact, the nail behaves just like a bar magnet."
http://science.howstuffworks.com/electromagnet4.htm

Notice the diagram at the bottom of this page which shows the nail
having a North and South pole (like a regular magnet).

As electrons (or in very simple terms the electricity from the
battery) flow through the wire they produce a magnetic field, the
direction of this magnetic field is dependent on the direction the
electrons travel.  Therefore if you reverse the connections the
direction of the magnetic field will be reversed.

"It [the polarity of the magnetic pole at the end of the bolt] will be
either a North or a South magnetic pole, depending on how it is
connected to the output of the diode bridge."
http://my.execpc.com/~rhoadley/magdcem.htm

"Don't worry about which end of the wire you attach to the positive
terminal of the battery and which one you attach to the negative
terminal. Your magnet will work just as well either way. What will
change is your magnet's polarity. One end of your magnet will be its
north pole and the other end will be its south pole. Reversing the way
the battery is connected will reverse the poles of your
electromagnet."
http://education.jlab.org/qa/electromagnet.html
 
I hope this answers your queries on this subject.  Should you require
any further help please ask for clarification and I will do my best to
respond swiftly.

Other useful pages:
://www.google.co.uk/search?q=define:Electromagnet
http://en.wikipedia.org/wiki/Electromagnet
http://www.energyquest.ca.gov/projects/electromagnet.html