The most common explanation, usually from those with military experience, is that it is a form of IFF - identification, friend or foe.
When Ukraine was part of the USSR, it had substantial manufacturing capacity, not least in terms of military vehicles and equipment.
After the breakup, Ukraine found itself with both factories and a large amount of now ex-Soviet inventory.
Since then it has manufactured its own military equipment but their ancestry is usually quite obvious.
To that end, with both Russia and Ukraine meeting on the battleground, they would be using equipment, that to the layman's eye, or to civilians, to those untrained, and to those without time to confirm identity, looked identical to each other.
In some ways the only other way to identify would have been camouflage painted on but in fog of war that might not always work as well. This is further complicated by the fact that some Russian units were seen with very similar camouflage.
So to that end, Russian vehicles were daubed with Z on them to identify them as friendly to their own forces.
Note that it was done last minute (some reports 21 Feb), just before rolling in, so that the OPFOR would not be aware of them or find out their meaning well before.
Example: Both countries use variants of the bronyetransportyor, armored transporter:

https://www.dialog.ua/ukr/war/246774_1645441351
Since they were first written about the markings and their meaning has had lots of variations seen, and the most copy pasted meaning version is this:
https://twitter.com/RuslanLeviev/status/1497216201616936964?s=20&t=6u1E6Xwood4CvzVuJXZRlw
Z - ZVO (ZVO), V - VVO (VVO), △ - SVO, / - most likely CVO.
Zorro is no longer Zorro.
or
https://zaxid.net/ukrayinski_viyskovi_pokazali_yak_vorog_markuye_svoyu_tehniku_n1537191
"Z" is the symbol for the Russian armed forces in the east, who are involved in operations in the Donetsk region,the
"Z" in a square or circle identifies Russian vehicles coming from Crimea,
the circle or "O" identifies vehicles from Belarus - some interpretations suggest that these are Belarusian vehicles, but this is not confirmed,
the "V" is a symbol for Marines - other interpretations indicate armed forces from the Brest region of Belarus,the symbol
"X" is used by the Chechens,
"A" is the symbol of special operations forces (SPETSNAZ).

(When the markings were first spotted before the invasion, some observers thought it representative of pre-invasion markings, meaning: 'preparations completed'}
Other common theories:
https://www.marca.com/en/lifestyle/world-news/2022/03/01/621e58eee2704e3e4f8b45f7.html
"Often these symbols will be location-based - they will communicate where the unit is going," Michael Clarke, former director of defence of the think tank RUSI, told Sky.
"They are probably signals that tell which units are heading to the northeast or northwest of a district, for example."
https://www.hitc.com/en-gb/2022/03/07/what-does-the-z-on-russian-tanks-mean/
Without official confirmation, there is only speculation behind what "Z" could mean. Kamil Galeev, a former Galina Starovoitova Fellow at policy think tank The Wilson Center, tweeted that some interpreted the "Z" as short for "za pobedy" — the Russian term for "victory." Others have guessed the "Z" is short for "zapad" (or west) and is meant to signify west-bound infantry.
https://fortune.com/2022/03/07/russia-z-tank-marking-how-letter-became-symbol-pro-ukraine-war-invasion/
How Z came to be
Experts suggest the symbol was first used as a unit identification marking to avoid friendly fire between Russians. Usage of the letter Z was first reported on Feb. 28 by U.S. armed forces defense journal Task & Purpose, which argued it was a way of distinguishing Russians from Ukrainians, as both sides use similar military equipment.
The journal notes that in the fog of war, “a Russian T-72 main battle tank may look a lot like a Ukrainian T-80 through long-range sights.”
The Russian Defense Ministry issued a statement on March 3, noting that Z stood for Za pobedu, or “For victory,” while V stood for Sila v pravde, or “Our strength is in truth.”
But the usage of more than one letter has left some believing the letter signifies the end location or mission of the vehicle. Former Marine Capt. Rob Lee, who spent a year with a defense-focused think tank in Moscow and is now a Ph.D. candidate at the Department of War Studies at King’s College London, notes the Z symbol is “different from what you normally see on Russian vehicles.”
“They’re obviously something new. And the most likely reason they would have put these kinds of symbols on is to indicate a different task force, a different echelon,” (Note: See later on for Russian/Soviet era formation markings, which could be related)
Since its introduction as a military fixture, the letter Z can now be found anywhere, from T-shirts to car stickers. Online usernames have capitalized the letter to show support, and even children in hospice were reportedly lined up to form a Z shape.
https://taskandpurpose.com/analysis/russian-military-equipment-white-markings/
Here’s what those mysterious white ‘Z’ markings on Russian military equipment may mean (FEB 24, 2022)
“Bottom line is the ‘Z’ markings (and others like it) are a deconfliction measure to help prevent fratricide, or friendly fire incidents,” said Air Force Lt. Col. Tyson Wetzel, senior Air Force fellow with The Atlantic Council think tank in Washington, D.C. Because Russian warplanes, such as SU-25 Frogfoots and SU-34 Fullbacks, fly too quickly for their pilots to recognize the “Z” markings, these symbols are more likely meant to deter fratricide from Russian attack helicopters, artillery, rocket launchers, and mortars, Wetzel said.
https://www.thedrive.com/the-war-zone/44413/mysterious-symbols-are-appearing-on-russian-military-vehicles-near-ukraine
Mysterious Symbols Are Appearing On Russian Military Vehicles Near Ukraine (FEBRUARY 22, 2022)
So far, photos and videos that have appeared on social media show a range of different symbols, the implication presumably being that these will be used for rapid identification on and around the battlefield. This is especially important when large numbers of vehicles are headed in different directions and require marshalling by troops or other security forces on the ground who may not be immediately familiar with their units and objectives.
https://jauns.lv/raksts/arzemes/488981-z-apli-un-citas-figuras-kas-ir-noslepumainie-simboli-kuri-rota-krievu-militaro-tehniku-pie-ukrainas
There are a great many other speculative meanings posited (WWZ, Zelensky, victory, point of no return, ready for invasion, etc) and most start with Z and do not count for the other letters seen.
Similar speculation:
Meanwhile, there are several communication groups on social networks, especially the Telegram, which bring together representatives and supporters of the separatist "republics" in Donbass, who are passionate about spreading the "Z" symbol as such a mysterious sign

..and there are occasional reports indicating that such markings were seen in Crimea in 2014 and in Syria.
I have not seen any such photos so unable to say, but it does remind me of this:
Russia's Use of Unmarked Troops in Crimea
Beginning in late February 2014, unmarked military personnel
suddenly appeared in Crimea and, alongside Crimean "self-defense"
forces, subsequently took control over key strategic and military
facilities in the peninsula. The unexpected appearance of these unmarked soldiers-referred to by some commentators as the "little green men scenario" -left both the Ukrainian authorities and western states puzzled about the affiliation of these troops.
https://scholarship.law.vanderbilt.edu/cgi/viewcontent.cgi?article=1213&context=vjtl
(Illegally Evading Attribution? Russia's Use of Unmarked Troops in Crimea and International Humanitarian Law)
Identifying the actors on the Ukrainian battlefields and outlining their legal responsibilities and obligations is not simply an academic exercise.
Further, crystallizing the battlefield status of the actors eliminates any later claims of ignorance, while simultaneously increasing the likelihood of accountability.
https://digital-commons.usnwc.edu/cgi/viewcontent.cgi?article=1321&context=ils
(The Combatant Status of the “Little Green Men” and Other Participants in the Ukraine Conflict)
..and so, if it is true such markings were also used in Crimea, it may have also been for IFF from their own side since the Russians in particular were lacking the standard identification features.
Maybe related:
Back in 2017, this was posted when looking at tactical markings of de-mothballed equipment turning up in the region:
https://informnapalm.org/en/valuyki-boguchar-common-markings-military-equipment-operated-russian-army-lpr/
And, harking back to Soviet vehicle markings:
At regimental level, formation identification markings, which seemed to be temporary but often left were spotted alongside vehicle numbers as well as formation symbols.
Consisting of simple shapes usually a circle, square or rectangle. The addition of smaller markers inside increased the availability of symbol variation.
http://army.armor.kiev.ua/hist/opoznav.shtml


Each division was assigned one specific geometric figure. Letters, numbers, dots, lines, sectors were applied inside the divisional sign, which denoted the regiment, a separate battalion of the division.
In the regiments, it was common practice to give numbers not in a row, but to encrypt them.
There were several more such encryption systems. Here everything depended on the imagination of the regiment's deputy commander (the regiment commander usually did not interfere in this himself).
As a rule, nothing else was required, and there were no other identification marks or numbers on the armored vehicles.
Which might itself give rise to the scarcity of confirmation about the actual meaning whilst the operation is continuing...

...........
As the other answer mentions, as time goes on the apparent meaning has evolved in its representation:
Recently it seems that these symbols, usually the Z, are being used for propaganda purposes:

- for our people
- for the victory
- our strength is that we represent the truth
- the objective will be completed
https://strana.today/news/379919-chto-oznachajut-bukva-z-i-v-na-voennoj-tekhnike-armii-rossii.html
Edit:
With regards to Zapad 2017:
Zapad's were large scale military exercises carried out by the former Soviet Union, roughly every four years, and most recently by Russia.
But like Vostok, Tsentr, and Kavkaz, these are named for the region of the country in which they are held.
As such each exercise has had focus on different aspects of military action (2009 was an attack against Poland), and both Zapad 2017 and Zapad 2021 for example was fought in an 'active' defense role, the latter being much larger.
https://www.europeanleadershipnetwork.org/commentary/the-political-logic-of-the-zapad-exercises/
https://www.themoscowtimes.com/2021/09/23/zapad-2021-what-we-learned-from-russias-massive-military-drills-a75127