[C]apital is not a thing, but a social relation between persons which is mediated through things.
The next time you feel intimidated by someone using abstruse economic jargon to explain contemporary events, just remember that although there were probably many fascinating technical details at work behind the institution of slavery in, say, the American South, about 99% of what of anyone really needs to know about it concerns the relations between people. If you leave out the relations between people, no amount of technical exposition is going to capture what was going on in that system.
One of the most satisfying aspects of Capital, Vol. 1 comes from Marx's insistence that we look at our system in the same way. While there is no shortage of technical arguments or challenging language, one can take comfort in knowing that no matter how complex the concepts become, Marx has only proceeded thus far by first posing the same question about capitalism that any sane person would ask about slavery: what does it actually mean for the people involved?
One of the ways Marx does this is by defining capital as a social relation, not a thing. Capital, and the economic system named after it, implies particular relations between people, just as slavery implies one group of people relating to another group in a specific way. It is the relations between people which define the logic of the system, not mathematical formulae. This is frequently lost on Marxists and non-Marxists alike.
If we only understand capital as a thing, we too easily miss what really matters, because the whole conversation turns into a one-sided story about the entrepreneur, this person who takes "capital" and does all kinds of wonderful things with it, like creating wealth and jobs. That's probably 100% of what you have heard. Marx is saying: you might as well talk about all the good things the slave owner does, like provide for his family or give money to his church or allot food and shelter for his slaves, etc.; because the fact that wealth is created, or that people are recruited into the process which creates it -- these qualities don't distinguish capitalism from slave societies, or any social system that came before. So what distinguishes capitalism specifically?
The answer always comes back to what is happening between people.