Socialists want to eliminate Private Property, which is to say property that generates capital: farms, factories, stores, etc. While retaining personal property. Stuff like you're house, your tv, your toothbrush, etc.

Now, there are some exceptions to this. If you go to Cuba, you'd find that most of the agricultural land isn't collectivised, but is rather owned by individuals or families, who's interests are represented by a mass organization of small holding farmers.

The historical specificities of Cuba's land reform process meant that Cuban socialism retained a sector of small scale private ownership, within a Socialist system.

The point Marx is making is that under capitalism, as much as they talk about private property being a fundamental natural right, the consolidation of capital means that most people never have the opportunity to access that supposed right, or reap its benefits