Capitalism is freedom applied to economics.
That means no one forces you to do anything or interferes with you. If you want to trade or buy or sell, it's between you and the guy you trade with, and no on else. The only condition is that you both agree to the trade. Trades must be voluntary.
This is a good system because it means all trades are for the mutual benefit of all the parties involved. People offer to trade when they see a benefit for themselves, and trades happen when everyone involved sees a benefit for themselves. Capitalism consists of win-win trades (each person wins according to their best judgment).
All alternative systems to capitalism involve changing this basic formula in some way. That means less freedom, restrictions on some win-win trades, involuntary trades, trades where someone loses, or something like that.
For example, maybe some trades are made illegal. That means you have less freedom, and you may be prevented from making a trade that would benefit you, and benefit your trading partner. In that case, all trades that happen are still voluntary, win-win trades.
Or maybe the Government decides you can make all trades, but only if you pay a tax for each trade. In this case billions of possible voluntary trades for mutual benefit become illegal, and in return people are offered the chance to make a different trade involving one more party. The new party is given money, and adds nothing to the trade directly (the third party offers a generic list of services to everyone, whether they trade or not). This makes the trade worse for the original people involved, and better for the Government. Trades will still happen when the benefit to both parties is large enough that they still benefit even after the Government takes some money. Trades for smaller amounts of benefit will be prevented. People are not free to trade without involving the Government in this way.
Communism takes from productive people and gives to unproductive or needy people. This involves involuntary transactions. I'm not free to keep all my stuff and only engage in trades I believe benefit myself. Instead, sometimes my stuff is taken and I get nothing for it, or something I value less.
Capitalism is difficult to argue against. If I want to trade, and he wants to trade, who could object to our voluntary trade?
There are a few common objections. The first is that maybe he's paying with fake money or selling shoddy merchandise. Something has to be done about fraud. Of course I agree. Capitalism (and all other systems) fares much better when you also have law and order to prevent force and fraud.
Another objection is that I might be selling weapons to a criminal who is going to go kill people. But negligently risking participating in murders is a crime. Exactly what constitutes participating in a crime, and what kind of safeguards are required, is a tricky subject. Capitalism has no position on this matter: it is neutral.
Another objection is that we need taxes to pay for the Government, which we need to provide services like the police and army. First of all, capitalism with reasonably low taxes is still a pretty good system. It doesn't ruin our lives to pay taxes. But there is room for improvement. Something less than ideal is definitely going on when many voluntary and beneficial trades are prevented by force. If we could replace Government with something completely voluntary, that is still effective at the important functions of Government, that'd be better.
Improvement can take various forms. The army could be run more cost efficiently, and then taxes could be lowered; that'd be progress. Or some Government services could be run by private, for-profit companies. There's no reason it's impossible to provide valuable services and get people to voluntarily pay for them. It may be tricky or difficult sometimes, but when you're providing value it's definitely possible to get paid somehow or other. Once a private company is providing a service well, then it's no longer necessary for the Government to provide it, and it can be discontinued. Charities can also provide services. If a service is valuable and important, and everyone wants it, why not voluntary donate money for it instead of paying taxes? This may be tricky to set up, but there's no fundamental reason it'd be impossible, and once it is set up and working well then Government could shrink. The theme here is that there is room for improvement in making the Government smaller and smaller as we create superior alternatives, and in doing so we'll have a more capitalistic, more voluntary, and more free society.
What if I can't afford a service that the Government used to give away for free? Well, why do you deserve to have free services at other people's expense? Why should they work for your benefit? Why do they owe you this service? Wanting to intrude on other people's lives, especially to demand things of them, is a nasty attitude. That said, in some cases it may be good that you have this service for free. If so, explain to someone why it is good, and he will help you. If you can't convince him, try someone else. If you can't convince anyone, then why do you think you're right and everyone else is wrong? It's still possible you are right. But there are no systems which always do the right thing no matter what. That's just wishing for utopia. A good way to think about it is this: can the use of force or involuntary mechanisms make the right thing happen more often? It only could if you assume the people in power are usually right which is an unreasonable premise to design a society around.
What does it take to be correct in most debates? It doesn't take being powerful. Having a lot of guns has no bearing on the truth. What it takes is rational thinking, imaginative guesses, and ample criticism. We can best support these traits by letting people think for themselves and make their own decisions, and not expecting them to do anything we can't persuade them of. Further, what if the people in power have a mistaken idea? How will it be corrected? They don't treat disagreement in a respectful and rational way — instead they use force — which prevents them from learning much. Or if they use force only sometimes, then they sabotage their learning sometimes, which is bad too, and the badness matches the times they deviate from voluntary interactions.
Consider when force will be used. It will never be used when the people in power have a persuasive argument. They won't use force when they could get their way with words alone. At the least, words are much cheaper than force, and will maintain a better reputation for them. Instead, force will be used in exactly the cases where their argument is weak. Force will be used when they cannot persuade people. They pretend to back up their decisions with force because they are wise, but in fact they do it because their reasoning is weak. So a system that uses force, at all, will be less rational and make less good decisions. (This argument is due to William Godwin.)
Because it makes freedom and voluntary interaction its fundamental principles, capitalism is a rational system under which people can thrive.