Does Thanos intend to exert his plan on each planet individually or the entire universe as a whole?

Based upon how Thanos explains it, planet by planet is most plausible

Thanos explicitly explains his motivations in the film. He's trying to to spare other populated worlds from the fate that befell his. Which is to say, collapse due to overpopulation and competition for/exhaustion of finite resources.

Prior to acquiring the infinity stones, his method for accomplishing that goal was to visit every planet and then use conventional measures to eliminate half of its population. Whether 50% is some ideal number for reducing resource contention doesn't matter, what matters is that Thanos seems to have considered it to be. He could have just as easily killed 20% of the population, or 70%, or anything else, but he settled on 50% as the "correct" proportion for his goal.

So the 50% per planet seems significant, and although killing 50% of all the people on each planet will produce the same bodycount as killing 50% of all people in the universe, there are statistical differences between the approaches.

A demonstration probably works best, so here's a Thanos Murder Simulator (or an alternate blue/orange version).

Notably, killing 50% of the people in the entire universe produces a result like this (each box is a person; green is alive, red is dead, groupings of boxes are planets):

Notably, one unlucky planet has only 6 survivors (nearly a 75% kill rate). A couple of others have lost only ~25% of their populations. If you need to kill 50% to curtail overpopulation on a planet, then killing everyone in the universe at random is a bad way to do it. You'll get some planets emerging relatively unscathed, and others catastrophically depopulated (I mean, even more catastrophically than 50% would be).

You get a much more uniform result by killing 50% of the population on each planet:

Considering Thanos's stated goals/motivations and the way he went about culling planets before completing the Infinity Gauntlet, the latter scenario seems like what he would go for.

Other characters say 'half the universe'

However, these other characters are not Thanos, and may be paraphrasing for expediency.

The only character who knows what Thanos intends is Thanos, and he gets to deliver a fairly lengthy explanation in the film. His explanation leans more towards randomly killing 50% of the population of each planet than it does towards 50% of everyone in the universe.

The bodycount is the same either way, but to Thanos it's the distribution of the bodies that's most important.

Using a Binomial distribution with number 7.5 billion and probability 0.50, the chances of Earth's death count from a universal unbiased killing resulting outside of 49.99%-50.01% of the total population is 1.7*10^-132, which is less likely than flipping 438 heads in a row with a fair coin.

_{Updated: I was first using a Poisson distribution which is a limiting case of a Binomial distribution (old claim: The chance of >50.01% (or <49.99%) of Earth's population being killed universally randomly is 3*10^-32! This is less likely than flipping 100 heads in a row with a fair coin). I found this site now which can handle the very small numbers involved in the use of the error function.}

So, it doesn't really matter which interpretation is the true case: Planets with realistic populations and even very accurate census won't notice the difference.

Full disclosure: I haven't seen the movie, nor have I ever read a comic book or know who any of these characters are. But I'm familiar with statistics and probabilities and have a habit of browsing the network Hot Questions, so...

Explanation: The Binomial distribution is useful to see the results here. It gives the probability of observing integer random outcomes (like number of people killed) from any expected average (like half of a population). My use depends on the following assumptions.

• The gauntlet will kill exactly half of the universe's (sentient?) lives (rounded if total is odd).

• The gauntlet will take absolutely nothing into account -- the chosen to die are chosen randomly to meet this quota. This means the gantlet does not travel planet-to-planet, or choose those to die one at a time, or kill the oldest, the smallest, the smelliest, etc.

• The population of any planet is a small fraction of the universe's population. If Earth comprised all of the universe, obviously exactly half of us would die. But if 10 googolplex exist, and thus 5 googolplex must die, each of our deaths makes a negligible impact on the quota and our chances of living are unchanged -- 50%.

• The Marvel universe has no more than a trillion trillion trillion times as many galaxies as our own universe, and each of its stars has no more than a trillion populated planets each. These very large (and very small) numbers can be hard to wrap your head around, but no scientific estimate of the number of galaxies/stars/planets in this universe comes close to legitimizing the chances of the asymmetric planet-wise killings proposed in the comments/answers.

The Math: From the binomial wiki page linked, the standard deviation is $\sqrt{np(1-p)}$, with p=0.50. For the Earth (n=7.5 billion), this is 43,301 people. In terms of sigma, 0.01% of our population (7.5 billion) is 17.32 sigma. As a rule of thumb for statisticians, there is a 99.7% chance of drawing a number from within three sigma from the mean. Likewise, there is a 99.999999...% chance of drawing from within 17.32 sigma from the mean. Put another way, with the error function tool linked above, the chance of drawing outside of 17.32 sigma from the mean is less than 1 in 10^100!

The range of likely death counts from this event is much smaller than Earth's census would even notice. Moreover, as others have explained, the "kill randomly universally" method also avoids the complication of identifying to which planet (or ship, or rock, or floating naked in space...) each person belongs.

It was mentioned that Thanos wants to kill half the population of the universe.

From transcript of Avengers: Infinity War,

Gamora: He won't stop. Until he destroys half the universe. Everything you know. Everything you love. It will all be gone.

However, before acquiring the Infinity Stones, he actually went to each planet and kill half the population there. This part might have created the confusion.

From transcript,

Gamora: No, no, we were happy on my home planet.

Thanos: Going to bed hungry, scrounging for scraps? Your planet was on the brink of collapse. I was the one who stopped that. You know what's happened since then? The children born have known nothing but full bellies and clear skies. It's a paradise.

Gamora: Because you murdered half the planet!

and

Thor: There's six stones out there. Thanos already has the Power Stone because he stole it last week, when he decimated Xandar. He stole the Space Stone from me, when he destroyed my ship and slaughtered half my people. The Time and Mind Stones are safe on Earth. They're with the Avengers.

Also, different sources like this and this mention that he wants to kill half the universe.

Killing half population of the universe and half population from each planet seem similar if we do the math. If he kills half population from each planet, he's still killing the half population of the universe.

Thanos tells Tony that when he has accomplished his goal, half of humanity will remain. This is important, because he we know that he intends to choose his victims randomly.

If Thanos intended to simply draw his victims from one big pool containing the entire universe, then he could not be certain ahead of time that half of humanity would remain. By random chance, he could very well draw more of Earth's population than that, or less. It's even possible, if unlikely, that he could wipe out Earth entirely, or leave it completely unscathed.

Let us suppose that Thanos got a big bag of marbles -one for each person in the universe, half of them red, half of them blue- shook it up, and forced everybody to draw. Everyone who draws a red marble disintegrates. This is guaranteed to kill exactly half the people in the universe, while still being random. However, with trillions if not quadrillions of marbles in the bag, there are going to be runs of the same color that last for thousands, millions, maybe even billions long. It's entirely possible that everyone on Earth might draw red marbles (i.e. Earth is wiped out) or blue ones (i.e. Earth is unscathed). Earth probably wouldn't even be the only world this happened to. The universe is a big place.

But that's not how Thanos acts. He knows that half of humanity will survive. The only way he can be sure of that, while still holding to his idea of "fairness through randomness", is if he has another method for choosing his victims. He would need to divide them up according to their worlds, and choose half the people on each: in other words, a bag of marbles for each planet, rather than one for the whole universe. This is similar to the method he used before he had the Stones, though faster and more efficient. More to the point, it's more closely in line with his idea that the resources on each world need to be distributed to half as many people: he needs to be sure that there are actually half as many people on each world, and this requires a finer-grained approach than throwing everyone into one large pool.

Thanos intends to kill half the population from the whole universe, randomly.

My impression while seeing the movie was that he meant planet by planet, although this was likely caused by seeing Thanos visit each planet and killing half the population manually before he had the gauntlet.

However, if we look into the scene where Thanos snaps his fingers we can see on Titan that this was not the case. At the time of the snap, Titan had 7 people. Iron Man, Spider-Man, Dr. Strange, Star Lord, Drax, Mantis, and Nebula. The snap killed 5 people. If his snap killed half of each planet, we would expect to see only 3-4 people dead.

My conclusion says this is done on a universal level, rather than a planetary one.

I think both of your statements are false. Why? Let me 'splain...

We know that the 'snap' wasn't localized to Earth because characters on Titan also flaked away so he's not going planet to planet.

We also know that, on Titan, about half of the Human characters flaked away (Starlord, Dr. String and Spider-Man)... There were an odd number of humans so you couldn't get EXACTLY half. That would just be gross.

Therefore, this is a bit of a leap using available info from only the first IW but I believe the snap effected the Universe as a whole but NOT at random. I believe that half of all species on every planet were hit at the same time. Where there was only 1 of a specific species, (thor, rocket & groot on earth or drax & mantis on Titan), they could stay or could go based on whatever intent was behind the snap. Where there was an odd number, the "odd man out" could stay or go based on the same intent

I think the previous answers underestimate the power of the Infinity Stones and the ability to control intent across the universe when wielding them all.

Thanos WAS doing it on a planet by planet basis, manually and sequentially, since that was what was within his capabilities.

The whole point of gathering the Infinity Stones and wielding them with the Infinity Gauntlet was that he could take care of the whole universe, instantly ("snap of his fingers").

Since your question seems more focused on whether "randomly" translates to an entire population on a particular planet being untouched, potentially and an entire other planet getting wiped clean, using the entire universe as a random pool would kind of defeat his (to himself, at least) altruistic intentions. The idea is that the universe's resources are finite, but so are each planet's. In order to save populations from their own excesses, the "pruning" would have to happen both locally and universe-wide, to achieve his goals.

What's interesting is that on Titan people were taken out, but that appears to be within the pool of "Earth" population, in practice. It appears that he's going by each sentient species and planets they inhabit, not just visit.

Though I'm not sure that this is canon, because the movie makers might not have considered this at the process-design level.

Thanos intends to kill half the population from every planet, randomly.

Because the motive of killing was that the living beings will have more available resources and there will be less sharing. If he kills from every planet then the other half of the population will always have the resources. But if he kills randomly from universe, then, some planets will still have same number of beings and some might be wiped off clean with all the resources and nobody to use it.