On the negative side of one speaker, there are two cords with black
insulation. One goes to the plug-like thing that probably connected it
to the stereo's main board. The other goes to the negative side on the
other speaker (which accidentally got ripped out but I can probably
solder that back on.) That same speaker has a grey cord on the
positive side that goes to the plug. The other speaker has the
previously mentioned black cord on the negative side and a brownish-
reddish cord on the positive side that goes to the plug.

It sounds to me as if what you have here is a stereo speaker system.

The "two cords with black insulation" are the common (ground) side of
the speaker connection. It's a three-way connection, with the ground
contact on the plug going to one speaker and then another wire
continuing to the other speaker.

Each speaker then has an independent "hot" wire - one grey and the
other brownish-red, apparently. Each of the hot wires goes to an
individual contact on the plug.

This is very much the same sort of arrangement you'd see with a stereo
headphone setup. If you look at the plug for such a headphone
(usually a 1/8" mini "phone" plug) you'll see three contacts... the
tip (at the end) which is normally the left-channel "hot" connection,
the ring or collar (right-channel "hot") and the sleeve or base
(ground or common).

See http://en.wikipedia.org/wiki/TRS_connector

So my question is, why do regular headphones and speakers have only
two cords, and if I were to modify this for use as a regular speaker,
to plug into my mp3 player or something else, what would I do with the
extra cord, and which one is the extra cord?

There is no extra cord.

Each individual speaker needs two wires. If the signals (and
amplifiers) were independent and isolated, you'd need a total of four
wires to drive two speakers - two wires for the left speaker and two
wires for the right speaker.

Commonly, though, the amplifier is set up so that one "common" wire is
connected to the amplifier's ground, and can serve as the signal
return for both speakers. This eliminates the need for one extra
wire, and you can get away with three - one ground, one left-channel
hot, and one right-channel hot.

Monaural headphones only need two wires, because the left and right
channels are being fed with identical signals. That's not what you'd
want for an MP3 player.

See the Wikipedia article above - it should give you enough
information to wire this speaker up for normal stereo use with a 1/8"
miniphone plug (which is what most MP3 players use).

It is questionable, however, whether a standard MP3 player has an
amplifier powerful enough to drive a loudspeaker. Mostly, they're
designed to drive higher-impedance headphones (32 ohms or above) at
relatively lower power levels. Loudspeakers need more power to
drive... either an amplifier in the stereo, or (in the case of
computer A/V speakers and most external speakers for MP3 players) an
amplifier built into the speaker itself.

--
Dave Platt AE6EO
Friends of Jade Warrior home page: http://www.radagast.org/jade-warrior
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