For tax purposes "gross profit" is gross receipts (i.e. revenues from sales) less returns and allowances, and less costs of goods sold.
For example, if your business is a department store, gross receipts is the cash that you are paid for things you sell, returns is money paid to someone who bought something and then returns it because it doesn't fit or they don't like it, allowances is money paid to someone who bought something to compensate them for the fact that it is defective or flawed when the goods aren't returned to you, and costs of goods sold is what you paid some wholesale company to buy your inventory. In a manufacturing business, cost of goods sold would include the cost incurred to build the products you sell to wholesalers.
"Operating income" is gross profits less operating expenses (the vast majority of deductions).
For example, continuing the department store example, operating expenses would include salaries paid to workers at your stores, payroll taxes, worker's compensation and health insurance bills for those employees, rent paid to shopping malls, utility bills for your stores, cleaning supplies you buy to keep your stores clean, money you spend on labels and cash register supplies, credit card company merchant account fees incurred in connection with your sales, and advertising expenses.
"Net profit" is operating income, increased by investment income (from investments like bank account interest or rent on real estate not currently being used in your business that is not related to the operating primary business of the company) and reduced by non-operating expenses such as interest expenses (interest paid on corporate bonds is usually the biggest one in a publicly traded company).
While we're at it, "total income" is gross profit plus investment income, prior to operating expenses and non-operating expenses, and I think you can guess what "total expenses" means.
The contributing components aren't defined in precisely the same way for tax accounting purposes and for financial accounting purposes, but the basic concepts are similar. A quarterly report would be using the financial accounting definitions defined by Generally Accepted Accounting Principles (GAAP), which are defined by a non-profit sponsored by professional organizations for CPAs, rather than the tax definitions (see here).
One of the more important differences is that in tax accounting, net operating losses can be carried forward to future years, while for financial accounting purposes, operating losses are always taken in the year that they actually occur.
Of course, the real joy of it for an accountant comes in the definitions of these sub-terms. There are lengthy books and other accounting guidelines (at about three main levels of generality) and various kinds of tax authorities (statutes, treasury regulations, revenue rulings, revenue procedures, letter rulings, tax court cases, ordinary court cases, etc.) that fill in the details of these broad categories.