I've been playing civilisation for just over 5 months and I really don't know how to capture a city. In the industrial era, I invaded one of Poland's cities and easily knocked it down to no health! But the problem is the city wouldn't become mine! I don't know if there's a rule or something that says you have to capture their capital city to take over their civilisation. They have a artillery unit in their city and that's it. They have sea to the south and I am storming them from the west. Their capital city is slightly east of the city I am attacking and I have started sending troops down their. I am running low on soldiers and the only thing that is keeping their cit down is my subs blocking sea-control and reinforcements and my great war bombers and tri-planes! Also, when I scroll over their city with my mouse it says: 'If you take this city you will receive a major warmonger penalty!' I don't know what that means, but I'm guessing it's really bad (lose a lot of money or something). Please help!
You can capture a city simply by melee attacking it, like you would do with a normal unit. Remember that you can only take cities with a melee attack, you cannot take cities with ranged or air units. You can take cities with naval melee units, such as the Privateer or Destroyer.
To entirely wipe out another civilization, you will have to conquer all of their cities. But in order to win a domination victory, you only have to be in control of the capital cities of all civs in the game.
About the warmonger penalty; this is diplomatic penalty you will get with other civilizations in the game. This means that they will like you less, resulting in them giving you less money for luxury and strategic resources, or if it gets worse, they will denounce you or even declare war on you. Taking the last city of an enemy will result in a much larger penalty. You will not lose any money once you capture the city, if anything, you will gain income, but only after the city gets out of resistance. You will lose happiness once you conquer an enemy city, so make sure you don't drop too much into negative happiness.
I can also refer you to the Civilization Wikia page about city combat for more information.