Socialism means many different things to different people. Be very careful when talking to someone about socialism -- odds are that they'll think that you're talking about letting the commies march in and enslave us all. But that's not really what socialism is about...
The main idea behind socialism is that workers should not be oppressed by rich owners of capital (capitalists) who profit through the labor of the workers while doing little work themselves. While a government may do things to enforce this, such as protect workers from the more extreme cases of exploitation (child labor, slave labor, 80 hour work weeks, unpaid overtime, etc.), it's not likely to force all capital to be put under the control the workers (although in some centrally managed economies -- AKA communism -- the government justifies its control by saying that it represents the workers). The same goes for capitalism; the USA is for the most part capitalist, but the government does not demand it, and it would be hard (and pointless) to force all capital into the hands of profiteering individuals.
Various social welfare programs may also be used by governments to support the people in sickness, youth, and old age; this is sometimes considered an aspect of socialism. It should be noted that these could be run for profit (insurance companies do something like this), which would be consistent with capitalism.
A socialist company would be one in which everyone who works in the company is also a part owner, and everyone who owns part of the company works there. While 'capitalist' companies often find it useful to give its workers a personal interest in the company, they rarely give the employees any real power over how the companies' capital is used.
In Europe they have tried to set up some worker co-ops, but they do have some factors that may prevent them from being as competitive as traditional corporations. A socialist company may be slower to respond to a increase in demand, as the owners/workers may be reluctant to bring in new partners. There may also be a problem with deciding who to fire in a down cycle (who's going to vote to fire themselves?).
Both socialism and capitalism are compatible with a free market, at least in the sense that neither require a market regulated by a higher authority. A socialistic company might or might not work for the largest monetary profit, but both socialistic and capitalistic companies want to make a profit. There is nothing in either system to forbid taking full advantage of the demands of the market; the difference is that while a capitalistic company will (almost always) be pushed to make a profit at any cost (by its shareholders), the socialistic company will try to do whatever the workers in the company want (if the employees want to work twenty hours a day, sell at inflated prices, or work to become a monopoly, they can do it. More likely, they'll take just enough profit to buy nice cars and then take the rest of the week off. So to speak).
Addendum: Socialism is not against private property, either. Only the private ownership of capital beyond what the individual can use on his own. If you own a factory, you cannot hire wage slaves to work there; you can bring in other people to help with your work, but they would need to be given partial ownership in the factory, although this 'ownership' might be limited to the time they work there ('control' might be a better term than ownership in many situations).