What's the difference between the population mean of a variable, the distribution of sample means of a variable, and the mean of a variable?
Population mean: average calculated from a whole population.
Sample mean: average of a small sample of the population; approximates the population mean.
Mean: expected value of a random variable.
A population mean is the true (often impossible to know) average for the population. If you were able to measure a certain quantity for every single unit in a population and computed the average of all that data, that average would be your population mean.
Getting a population mean often would take way too much time and cost way too much money. There are also data that it is impossible to collect for an entire population, like the average lifespan of a AA alkaline battery. (Once a battery's life is measured, it cannot be used again.) Also, it is impossible to "finish" sampling a population when the data being collected change with each measurement, like a person's reaction time. For all these reasons and more, we settle for sample means.
A sample mean is an estimate of a population mean. It is computed by selecting a (fair) random sample of units from a population, measuring the quantity of interest for them, and computing the average of those data.
There are various methods for ensuring the sample mean is as representative of the population mean as possible, from splitting the population into categories and sampling equal proportions from each category, to grouping the population by regions and sampling every unit from a random selection of regions.
As a sample size increases (or as the sampling method improves), the distribution of the sample mean will be centred closer and closer to the population mean, and have smaller and smaller variance, meaning we should expect larger datasets (and better sampling methods) to give us closer guesses of the population mean, and smaller margins of error on those guesses.
The mean of a variable makes more sense in discussing probability rather than statistics. It is used when the population data is understood or can be known, like when we calculate the expected ("average") number of coin flips until we get 5 heads. (In this case, that would give a mean of 10.)