Socialism is a system where individuals do not own corporations. A lot of the other posters here are emphasizing the state ownership aspect where the government owns everything. But really socialism is about collections of people owning things without an elitist class (like modern billionaires). Who those collections of people are that owns things is where the interpretation and ambiguity begins though.

Imagine if Microsoft instead of having a board of directors and being on a stock exchange, instead was owned and operated by it’s employees and selected the CEO of their company by an internal election process, where the workers could select leaders that they thought would best represent their interests in the company. That would be another example of socialism that can exist without government involvement at all. The collection of owners here is simply the employees of Microsoft.