A company's balance sheet gives a picture of how all the assets, liabilities, and equities of the company "balance out."
The basic accounting equation is Total Assets = Total Liabilities + Equity, and a Balance Sheet is going to detail these parts to show how everything adds up at the time of the report.
With things equal on both sides of the equation, the company's books are balanced, the same way someone might go back through the carbon copies of checks they've written and "balance the checkbook" to make sure all checks written have been accounted for.
The way publicly traded businesses are structured, any asset balance over liability obligations is considered Equity, and generally most of this is going to be held in outstanding shares of Common Stock and Preferred Stock. Most companies are going to publish Annual Reports that contain this information, and they'll also file annually and quarterly with the SEC.
A balance sheet can be used to determine things like debt-to-equity ratio and acid-test ratio to get a picture of how healthy the company is, and whether it has the liquidity to pay its debts in a timely manner. Along with income statements and statements of cash flow, the balance sheet can be used for fundamental analysis to determine whether the company represents value when compared to peers in its industry.