It's nothing like that. It has nothing to do with the personal possessions that we all should be able to have. When that slogan was first used 'private property' referred only to private ownership of productive property.
It was - and still is - about denying anyone a 'right' to own factories, big farms and the means of distributing products, such as railways, airlines and road haulage fleets.
Anarchists are opposed to such private ownership because we are opposed to exploiting people. There are those, usually of the ruling class, who will deny that there is exploitation in the Ireland of the 1990s. All that stuff belongs to the bad old days ...or does it?
In the distant past things were a lot more obvious. A peasant had to work two or three days a week on the landlord's estate but got no payment for it. It was as clear as day that part of the fruit of that peasant's labour had been stolen by the lord.
Now workers are paid for all the hours they put in. Some may be underpaid by current standards, but they don't have to give their boss a set number of hours without pay. So how can anyone claim they are being exploited in the sense of having to work for nothing so that some parasite can benefit?
Under the present economic system - capitalism - goods are produced in order to be sold. Most of us do not have products to sell. We do, however, have something else to sell.
We have our ability to work, our labour power. Wages are the price we get for our labour power. Without labour power nothing can be produced. Even an apple on a tree has no value until it is picked, it is the labour used to pick it that gives it value. Otherwise it could not be eaten, it would just rot on the branch and be of no use to anyone.
It all seems simple and straight- forward. We work (if we are lucky enough to have a job), our work creates value,and we get paid for it. So what's the problem? It is that our wages never add up to the full value of our work.
The difference between what we get in wages and what the product or service is sold for (after allowing for expenses) is what bosses call profit. This is their source of income. This is the basis of capitalism, a small minority living off the unpaid wages of the majority.
Anarchists are working for a future where the ownership of industry will be taken away from the bosses and instead will become the property of society as a whole. Its control and management would be vested in bodies democratically elected by the workers themselves.
The world of work would not be geared to generating profits for a class of rich idlers like Tony O'Reilly, Margaret Heffernan or Michael Smurfit. Instead decisions about what to produce, and what to invest in improvements and new processes, would be taken on the basis of what is socially useful. Production would be geared to meet people's needs rather than to satisfy the greed of a tiny minority. That would be the end of 'private property'.