Market cap is what the market says the equity of the whole company is worth right now. It is the only sensible way to compare the size of two companies, because share price on its own says nothing: a fifty dollar stock with two billion shares is a hundred times larger than a fifty dollar stock with twenty million.
The conventional bands are large cap above ten billion dollars, mid cap between two and ten, small cap below two, and mega cap for the handful above two hundred billion. They matter because index membership, institutional mandates and liquidity all break along roughly those lines.
It measures the equity only. A company can have a modest market cap and an enormous debt load, so an acquirer would pay far more than the market cap to own it outright. Enterprise value is the figure that includes that.