I've been looking for things like the plugs displayed on these orange wires in this photo:
What are they called? I imagine that this is a part that I could buy somewhere.
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All they are are stiff peices of wire soldered to stranded Wire with a molded rubber case. you can buy them from adafruit, also called bread-boarding cables, jumper cables, (premium) wire jumpers.
OK, they are known as 'Jumper Wires' Here is a link to some that SparkFun sells. I'll bet that they can easily be located in different sizes, etc. elsewhere.
This page refers to them as 'leads' and gives some DIY instruction on making them stackable:
And again, for the DIY type person, on page 6 of this catalog is a crimp-style terminal.
Those look like single row breakaway headers, like these: http://rndwarehouse.com/breakaway-headers-1x40-straight-singlerow-extended-height-p-213.html?zenid=83og3qmqjb3g10obpv92sbavo7. I've done this in the past as it's the cheapest way to get non-solid core jumper wires for breadboarding.
In order to make them, you get a 40 pin header like the one I linked to, and then carefully break them apart, either with clippers or two pairs of pliers. You want to solder your stranded wire to the short end and then use shrink tube over the connection. The long end plugs into your breadboard.
EDIT -- looking at the picture again, they aren't breakaway headers, but I would still recommend going that route if you're okay with a little soldering.