In a sentence, stocks are a share of equity in the company, while bonds are a share of credit to the company.
When you buy one share of stock, you own a (typically infinitesimal) percentage of the company. You are usually entitled to a share of the profits of that company, and/or to participate in the business decisions of that company. A particular type of stock may or may not pay dividends, which is the primary way companies share profits with their stockholders (the other way is simply by increasing the company's share value by being successful and thus desirable to investors). A stock also may or may not allow you to vote on company business; you may hear about companies buying 20% or 30% "interests" in other companies; they own that percentage of the company, and their vote on company matters is given that same weight in the total voting pool. Typically, a company offers two levels of stocks: "Common" stock usually has voting rights attached, and may pay dividends. "Preferred" stock usually gives up the voting rights, but pays a higher dividend percentage (maybe double or triple that of common stock) and may have payment guarantees (if a promised dividend is missed in one quarter and then paid in the next, the preferred stockholders get their dividend for the past and present quarters before the common shareholders see a penny). Governments and non-profits are typically prohibited from selling their equity; if a government sold stock it would basically be taxing everyone and then paying back stockholders, while non-profit organizations have no profits to pay out as dividends.
Bonds, on the other hand, are a slice of the company's debt load. Think of bonds as kind of like a corporate credit card. When a company needs a lot of cash, it will sell bonds. A single bond may be worth $10, $100, or $1000, depending on the investor market being targeted. This is the amount the company will pay the bondholder at the end of the term of the bond. These bonds are bought by investors on the open market for less than their face value, and the company uses the cash it raises for whatever purpose it wants, before paying off the bondholders at term's end (usually by paying each bond at face value using money from a new package of bonds, in effect "rolling over" the debt to the next cycle, similar to you carrying a balance on your credit card). The difference between the cost and payoff is the "interest charge" on this slice of the loan, and can be expressed as a percentage of the purchase price over the remaining term of the bond, as its "yield" or "APY". For example, a bond worth $100 that was sold on Jan 1 for $85 and is due to be paid on Dec 31 of the same year has an APY of (15/85*100) = 17.65%. Typically, yields for highly-rated companies are more like 4-6%; a bond that would yield 17% is very risky and indicates a very low bond rating, so-called "junk status".