The main things that sometimes need to be calibratated:
(i) Head orientation (and possibly position) relative to the tape. Obviously, if the strip of tape on which "record head A" recorded signal and the strip of tape from which "play head B" plays it back aren't the same, there's a problem. Perhaps more important (and tougher to adjust): if "record head A" is twisted so the head gap isn't perfectly perpendicular to the tape, and "play head B" isn't twisted (or, worse, is twisted the opposite direction), at playback the signal at one edge of the track won't line up with the signal at the other, and you get cancellation of high frequencies. That's what "azimuth" is all about.
(ii) Bias. It's just a high frequency (way out of hearing range) signal that's mixed in with the audio ... basically to "loosen up" the electrons on the tape so they'll orient in accordance with the varying magnetic field that the heads put out (i.e. the signal you're trying to record). If the bias is off, you get distortion or weird frequency response. How much bias is right depends on the the characteristics of the stuff on the tape that you're trying to magnetize.
(iii) Basic electronic alignment: (a) the electronics will only behave properly if the voltage of the signal you're inputting is in a certain range and (b) the tape will only behave properly if the magnetic fields the heads hit it with are in a certain range. It's a good thing if these two ranges correspond ... so that, say, a signal that's at the high end of what can run through the electronics without distorting results in a magnetic signal that is more-or-less at the at the high end of what the tape can handle without distoring. This varies with the tape. It can get a bit fuzzy, because people like some types of distortion. Same concept, basically, at playback as at record. And, in order for your meters to be useful, their range also needs to correspond. So those are all things that might need to adjusted.
(iv) Record and playback EQ. The tape deck EQs the signal a bit when it records, and when it plays back. These have to match (more or less), or you'll wind up boosting or cutting varioius frequencies.
(v) Mechanical. If the motor turning the take-up reel pulls harder when there's more tape on the reel, the pull might be enough to pull the capstan so it turns faster, and the tape runs faster. That's an example. There are obviously a host of mechanical pieces that have to work together properly.
If you change to an entirely different tape formulation (like, say, from 456 to GP9), you may have to readjust all of bias, basic electronic alignment and EQ.
The other stuff isn't supposed to change, but over time it might. It's a good idea to check azimuth and the electronic stuff (basic alignment, EQ and bias) from time to time and adjust if necessary. I guess real pros check and tweak mechanical alignment periodically, though I suspect amateurs just worry about when it becomes apparent something's wrong.
