Capital is the means of production, minus land and labour. It's fairly comprehensive to think of it as all the tools, procedures, etc. that make up a business, whether that business is one guy in the yard of his hut or a multinational corporation. It's reasonable to think of a business itself as a machine, and the machine is capital.
Improving capital means making it work better. Designing and building better tools and procedures to produce whatever you're producing more efficiently. If you take some funny rocks, bake them in a pile of mud you carefully designed, and pound them into swords, you've improved capital. If you buy a hammer, an anvil, a forge and some iron, and learn to make horseshoes, you've improved capital. If you buy some guy's smithy and figure out you can make horseshoes cheaper by casting or stamping them, and build equipment and procedures to do that, then you've improved capital again. You started with existing capital, assembled it into a machine and/or innovated a bit and ended up with a machine (which is capital) that produces something valuable its original parts did not.
The most basic feature of capitalism is the idea that the benefit from doing those things should go to the person(s) responsible, and by doing so will encourage people to improve capital. Related ideas are that anybody who wants to should be able to engage in that process and, almost always, that a free market determines value.
That idea is opposed to, for example, feudalism, where improvements you make will primarily benefit the lord, or communism, where they benefit the group. You figure out how to make horsehoes better or make a field produce more and you maybe get a nice thanks from the boss man, but the same ration as before.
Capitalism existed long before corporations, particularly modern ones, legal liability, or anything like that. Many people identify capitalism with things that are distinctly non-capitalist because they're frustrated with aspects of "capitalism" that aren't capitalist at all. Corporations, for example, are typically very similar to fedual systems internally, and both employees and employers expect them to be.