I'm replacing a light switch.  I pulled out the old one, to find that
it was a GE switch, and the back of it was completely encased in
plastic, with only two black wires coming out of it - neither are
marked, and the actual wire is silver in color for both.

In the space where the switch was, It is impossible to distinguish the
color of the insulation, and there are 3 wires, all copper, and all
about 16 gauge, I estimate.  I have no idea how to tell which wire is
Hot, Neutral, or ground, so I can connect them to the right places on
my new lightswitch.

The other confusing part is that because the old switch had just 2
wires, whomever installed it just twisted 2 of the copper wires
together and connected them to one of the black wires, and connected
the remaining copper wire to the remaining black wire.

I have a multimeter, but not any clue as to how to use it, and I don't
want to get electrocuted.  It seems like a simple problem I should be
able to fix myself - I feel like getting an electrician is a waste. 
I'm aware of turning off the power at the breaker before doing any
work, but doesn't it need to be on for the multimeter to read
anything?  I don't even know how to use the positive and negative
leads off the multimeter to get a reading off of anything...  I've
searched for a while, then decided it might be best to get a specific
answer to my specific problem....

I'm assuming this is a simple single-pole switch (i.e. it's the only
switch that controls the light -not a 3-way or 4-way switch, where
more than one switch controls the same light).

Given that, a light switch only needs two wires - the "hot" and the
"switch leg", or just "leg".  The neutral wire is not switched, and
thus is often not in the same box as the switch.  The hot wire comes
from the electrical panel or circuit, and the switch connects or
disconnects this incoming wire from the leg wire, which is the wire
that goes to the light.  The electricity flows from the panel, through
the switch, coming in on the hot wire, and leaving on the leg wire,
through the light, and back via the neutral to the panel.  (Note:  of
course this is AC, so this isn't what literally happens, but is
illustrative.)  The neutral often isn't in the box with the switch,
since it doesn't need to be switched.

You don't need to know which wire is the hot or the leg to wire up a
switch - it doesn't matter- the switch will break the connection
either way.  The two wire twisted together are probably to carry it on
to another box.  The two wires twisted together could be the hot, in
which case the electrician probably needed to continue the hot on to
another box for another light somewhere else, or or the leg, which
they might do if there is more than one light being switched by that
same switch (in which case the two legs would each go to a light). 
Just leave the wires twisted together.

You should connect the new switch exactly like the old one was - one
screw terminal gets the single wire from the old switch, and the other
screw terminal gets the other two wires that are twisted together.  It
doesn't matter which terminal gets which one (except for the green
ground screw - see below).

**** I skipped grounding issues - this complicates things a little.  A
modern switch actually has three screw terminals - the hot, leg, and
ground.  The ground is usually on the bottom, and has a green screw. 
It sounds like your old switch didn't have a ground connection, and it
sounds like there is no ground wire in your box.  If the box in the
wall is metal, it may be grounded, in which case you can ground the
switch to that.  If not, you'll either have to leave it ungrounded
(likely not kosher with local electrical codes, but you'll be no worse
off than the old switch), or run a ground wire to it (likely difficult
to do).  In any case, don't connect a wire that's NOT ground to the
green ground screw!

If you want to use the multimeter, the idea is to turn the breaker
off, remove the switch, and arrange the wires so they're sticking out
of the box, and not touching anything.  Then you turn on the power and
use the multimeter, being careful not to short anything, or
electrocute yourself!

You can find better explanations on the web - try googling "light
switch wiring", etc. for some pictures.

To figure out which wire is the hot wire, you can use a "voltage
sensor" tool.  These cost less than $10 and can be purchased at
various hardware stores or at places like Radio Shack (search for item
22-106 at radioshack.com).

The tools have a light that illuminates when the tip is held within an
inch or so of a live wire.  They detect the electric field surrounding
A/C wires, which means that they don't need to physically touch the
wires you're testing, and that they work even when there's not a
complete circuit.  These can be used to find breaks in long runs of
extension cords (or theatrical power cables) as well.  They're often
sold as Christmas light testers since they can be used to quickly
figure out which light has burned out even if the entire strand has
gone dark.