What is the difference between the transition frequency and unity-gain frequency? I hear these terms being used all the time and I thought they referred to the same thing (the frequency at which the gain is unity) but I want to make sure I am not misunderstanding anything.
Well, you're on the right track.
Yes, the unity gain frequency is useful but today it's normal to talk of the Gain Bandwidth Product for op-amps. To take a modern example, the well specced LME49720, GBWP is stated as 55MHz. Rather better than the ancient' LM4558's 1MHz ! When the gain is 1 (unity gain frequency) the frequency is 55MHz. For almost every op-amp the gain at HF beyond the 'turnover frequency' (the frequency at which the open loop gain starts to reduce) reduces by 6dB/octave or 20dB /decade (dominant pole compensation).
fT (the transition frequency) is normally found in the data for discrete transistors. It's defined as the point at which the current gain (in emitter follower mode) falls to 1.
Modern audio power output devices, for example, have fT in the range 10-50 MHz. Many general purpose transistors have an fT of 100MHz. Switching transistors are more commonly specced by rise, storage and fall times. RF devices are another kettle of fish entirely !
'Old' single diffused power transistors had fT's as low as 800kHz ! e.g. 2N3055.
Clearly these concepts are interchangeable and no-one would berate you for using one characteristic over another.