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TCXO stands for Thermally Compensated Cystal Oscillator.
It is a very stable master oscillator so that the frequencies indicated by the radio's display are acurate and stable. Most TCXO's have worst case specifications around 1 ppm, some better, some a little worse. That means 1 Part Per Million, that means the stability is better than 14 Hertz at 14 MHz (and 7 Hz at 7 MHz). Radios without a TCXO may have specifications around 10 PPM.
Nowadays, with over 95% of all the ham rigs on the air being controlled by a master crystal oscillator (just my guess, I have no idea :-), you will find that most QSO's take place on rounded frequencies (such as 14.200 or 14.195 for instance) even though there is no requirement to be tuned to a round frequency in the amateur service. It is convenient and satisfying to see your rig display a round frequency during a QSO, so you can write that frequency in the log book (or if you use a software program for logging, the log will show a round frequency), but there is not much benefit beyond that.
My FT-1000 has a TCXO, and I cannot detect a frequency change as the radio warms up. An easy test is to tune to WWV signals, they transmit very stable and acurate time signals at 5 MHz, 10 MHz and 15 MHz, so that any time of day or night, you can hear at least one of those from anywhere within the continental US. Tune slightly off the frequency so you hear a tone while the radio warms up. If you cannot detect a change in the tone during the first 10 minutes, you are fine.
My FT-817 does not have a TCXO but it does not drift at all either. The FT-817 is a very recent rig.
With my TS-440, which does not have a TCXO (it has a normal, not thermally compensated crystal oscillator), the frequency drifts about 100 Hz (at 14 MHz) during the first 30 minutes of operation after the radio is turned on, after that, it does not move as long as the radio stays on. It has never posed a problem.
It is a nice feature, but there are few cases where you actually NEED it. One example of a case where you would NEED it would be if you want to pull out of the noise very weak signals using software such as WSJT, which is used for moon-bounce and meteor scatter on VHF and UHF. Another would be for Lowfers, hams who investigate the very low frequencies (below 1 MHz) where software is used to detect very weak signals burried into a lot of noise. In either case, the software used requires extremely good stability over long periods of time.
For all "normal" ham applications such as SSB and CW, even PSK31 or RTTY, you do not NEED a TCXO, even though it is nice not to have to worry about your radio drifting.