The below notes capture my understanding of the different nuances in the definition of UT1, TAI and UTC times.
UT1 time (Universal Time) is also known as astronomical time or solar time and is determined by the position of the Sun relative to the observer; as such it is influenced by vagaries in the Earth's rotation and does not flow uniformly. On a given meridian, when the Sun is at its zenith, it's 12 o' clock noon, by definition. This is the essence of astronomical, or solar time.
Given the various subtleties and physical world complexities governing the Earth's rotation, the exact duration of a UT1 day is not always the same. It therefore also follows that the exact duration of a UT1 hour, minute or second (understood as fractions of a UT1 day) are not the same either. This is simply saying again (with different words) that UT1 does not flow uniformly.
Also note that even though UT1 tracks the mean solar day, it is actually measured using distant X-ray sources, i.e. it is actually measuring the sidereal day and subsequently adjusted with a mathematical formula to arrive at the solar day.
TAI time (International Atomic Time) is a time scale that combines the output of some 400 highly precise atomic clocks worldwide, and provides the exact speed for our clocks to tick. TAI flows uniformly by design but is unconnected to any astronomical observations.
By combining TAI and UT1 we get UTC (Coordinated Universal Time). UTC, like TAI (and unlike UT1) flows uniformly at a constant "rate" (driven by TAI). Given that this restriction (i.e. of uniform, constant flow) causes UTC to diverge from the solar time (which is tied to physical realities and thus "dirtier"), it has to be adjusted once or twice a year with the addition or removal of a second. Given that the angular speed of the Earth's rotation is slowing down over time, this means that TAI is getting slowly ahead of UT1 which is precisely why so far seconds have always been added, not removed from UTC. Note that this may sound counter-intuitive but the addition of a leap second is actually slowing down UTC to bring it more inline with UT1. Leap seconds are always added as the last second of a day so once or twice a year a UTC day had 86401 UTC seconds instead of 86400. But note that the duration of a UTC second is always identical to the duration of a TAI second.
On the subject of the difference between UTC and UT1, Wikipedia has this to say:
The time correction DUT1 (sometimes also written DUT) is the difference between Universal Time (UT1), which is defined by Earth's rotation, and Coordinated Universal Time (UTC), which is defined by a network of precision atomic clocks.DUT1 = UT1 − UTCUTC is maintained via leap seconds, such that DUT1 remains within the range:−0.9 s < DUT1 < +0.9 s.
Essentially, UTC is a compromise devised to satisfy the needs of two communities of users:
Pretty much the same is stated (much more clearly and eloquantly) in this physics.stackexchange answer (slightly edited):
UT1 (Universal Time 1) measures the Earth's rotation with respect to the distant stars (quasars, nowadays), scaled by a factor of (one mean solar day)/(one sidereal day), with small adjustments for polar motion. There are exactly 86400 UT1 seconds in a UT1 day.
TAI (International Atomic Time), which you didn't ask about, measures time according to a number of atomic clocks. There are exactly 86400 TAI seconds in a TAI day. The TAI second is based on the Earth's average rotation rate between 1750 and 1892. A UT1 day is now a couple of milliseconds longer (on average) than is a TAI day thanks to the slowing of the Earth's rotation rate.
This creates a fundamental problem. It is now universally agreed that atomic clocks provide a much better measure of time than does the Earth's rotation, yet for human comfort, we would still like time to stay in sync with the Earth's rotation. How to accomplish this?
GMT (an archaic term that is deprecated except in Great Britain) has now become another just another name for the time zone UTC+0h. Prior to 1972, GMT was the de facto standard that attempted to keep universal time and atomic time in sync. The old GMT adjusted the length of the last minute of every day to keep the two disparate concepts of time in sync. The BBC incorporated these daily adjustments in their broadcasts of the "six pips". The US and Canada did much the same with their radio-based time broadcasts. The daily adjustments used in GMT were becoming ever more problematic with an ever more connected and ever more precise world. These problems motivated the replacement of GMT with UTC.
UTC (Coordinated Universal Time) is the modern successor to GMT. A UTC second is by definition always exactly equal to one TAI second, but like the old GMT, a UTC day is not necessarily 86400 seconds long. The difference between the old GMT and UTC is that in lieu of the small daily adjustments used in the now deprecated GMT, the adjustments to UTC are infrequent and are always exactly one TAI second. These are leap seconds. The predictability and the current close match between UT1 and TAI means that leap seconds can be announced well in advance, only have to occur on June 30 or December 31, but can still keep UTC and UT1 within 0.8 seconds of one another. (Aside: This will not remain the case in the not too distant future.)
The rationale for switching to intermittent leap seconds was that doing so would keep time uniformly except for those intermittent leap second boundaries, and the problem introduced by those leap second boundaries could be easily circumvented due to the announcement of those jumps well in advance of when they would occur. This has turned out to not be the case; a number of different computer systems (e.g., Microsoft, Unix, and MacOS) have problems with leap seconds. There are moves afoot to get rid of leap seconds.
An insightful comment is found in the following paper:
Precision time and the rotation of the Earth, Dennis D. McCarthy Transits of Venus: New Views of the Solar System and Galaxy, Proceedings IAU Colloquium No. 196, 2004; c 2004 International Astronomical Union
… where the author says (in the conclusions section):
[we should] acknowledge that UT1 is essentially a measure of the Earth’s rotation angle to be determined astronomically, and improve the accessibility of this information to users by electronic means.… and also in the discussion section where the author comments in response to a question that:
Dennis McCarthy: Well, I didn’t have a chance to say it, but I think we need to get used to the fact that UT1 – UTC is just a way of expressing an angle, the Earth’s rotation angle. It’s no longer a time, UT1 is not really a time, it’s a way to express the Earth’s rotation angle and it should not be thought of as a real time scale.
… or "can somebody please blow my brains out"
See this horror story