Stocks (also commonly known as shares or equities) are investments in a company. Each stock represents a part (or share) of a company, so, by buying a stock you take partial ownership in a company and can therefore share in its profits (or losses).
Theoretically speaking, the value of a stock is determined by the present value of all future expected profits of the company and the share price will thus fluctuate as a function of how the company is expected to perform.
Equities may refer to regular stocks but may also be used as a term to refer to the other large sub-group of financial markets instruments, alongside fixed income. Taken in this sense, equities include regular stocks but also more complicated financial instruments such as equity derivatives.
