Stocks, shares, equities: different names, same thing. Americans tend to refer to stocks while we in the UK say shares. Shares give investors ownership of a company. They represent a portion of a company's assets and earnings. As an ordinary shareholder, the extent of your ownership in a company depends on the number of shares you own in relation to the total number of shares available.
Your shares allow you to attend the company Annual General Meeting (AGM), and sometimes to vote on important decisions to be taken by the company, where shareholders usually receive one vote per share.