I only remembered three things about the computer: it was a Tandy, it ran DOS, and the power button was red. I also know that my dad bought the machine a year or so before I was born, which meant that it must have been manufactured in 1987-ish.
Wikipedia told me that Tandy manufactured two lines of PC compatibles, the 1000 and the 2000. The 2000 began manufacture in 1981, which puts it slightly too early to be the model my dad bought. The Tandy 1000 began manufacture in '84, which is more likely.
However, there were quite a few variations on the 1000: the EX, the SX, the TX, the HX, and the SL and SL/2. The EX and HX subscribed to the "giant slab with a keyboard attached" school of design (similar to the Apple II), and I remembered the floppy drives being on the front of our machine. This would make it either the vanilla 1000 (unlikely since it would have been fairly outdated at the time of purchase), the SX, or the TX. (The SL's didn't begin manufacture until the year after I was born.)
It turns out that the vanilla 1000 had black floppy drives, whereas our machine was all beige, so that ruled that out. This left the SX and TX, which look virtually identical and were both first manufactured in 1986. I used Google Image Search to find a handful of pictures of each model of computer, and started searching for minute differences.
I eventually noticed the one visible difference between the two: the TX had a volume knob on the front, next to the joystick ports and the power button. I remembered the existence of that knob on our computer, mainly because it was a pain to turn because it was stiff and the edges of the little graspy handle thing were sharp.
So, I poked around a bit more in an effort to find some specs for the Tandy 1000 TX. Turns out that motherfucker had an Intel 80286 running at a whopping 8 MHz, 640KB of RAM, and MS-DOS 3.22.

It's pretty rad to be able to definitively add this to the list of Computers I Have Owned.