Let's start by defining what a bond is. A bond is essentially a loan. You loan a company (or a government) money and the company pays you back with interest. In the event of a bankruptcy, bondholders are paid back before stockholders.
You can purchase bonds in the secondary market, which means you are buying them from someone else rather than directly from the company. If a bond is trading at "par", it is trading exactly at what you would get paid if the company redeems (or pays back) the loan. For example, you own a bond that will be redeemed at $1,000 with a 5% coupon rate (or interest rate). If the bond is trading at par, it is trading for $1,000. If it is trading at $900, it is below par. Trading at $1,100 is above par.
The two major risks in owning bonds are interest rate risk and credit risk. Bonds are typically priced to reflect current interest rates. So if interest rates for comparable bonds are currently 6%, a bond with a 5% coupon rate would trade at less than par to compensate you for the lower coupon rate. So in general, bond prices drop when interest rates rise, and rise when interest rates fall. Credit risk is the risk that a company or government will become insolvent and you will not be paid back at maturity. So bonds with a higher perceived credit risk typically trade at higher interest rates to compensate investors for the higher risk.
Ignoring for the moment the risks in bonds, you should realize that bonds are an essential part of a portfolio. A typical investor owns a portfolio divided between stocks and bonds. Stocks represent more risk (but higher return), and bonds represent lower risk (and lower return). Setting the percentages devoted to stocks and bonds is a process called asset allocation. Asset allocation has been determined in studies to be the number one driver of the return of a portfolio, irrespective of the types of stocks or bonds owned.
The pros of owning bonds are that they dampen the risk in a portfolio and act as a hedge to your stock exposure. For example, if you are 60% stocks and 40% bonds, in the event of a stock market correction of 40%, your bond assets might drop quite a bit less, or even appreciate. That would give you an opportunity to rebalance your portfolio by selling bonds and buying stocks, perhaps the opposite of what the typical investor would do in a major market correction. This type of rebalancing as part of a well thought-out investment plan is an essential cornerstone of good investing.
The cons of owning bonds are that they are not the risk-free assets that many investors think they are. When you factor in interest rate and credit risk, you can lose money in bonds. Bonds can also be more expensive than stocks to purchase, and the dollar amounts to buy individual bonds are much greater than stocks, making it harder for smaller investors to buy a portfolio of individual bonds that is adequately diversified.
A great way to cheaply and effectively own a very diversified portfolio of bonds is to consider purchasing broad market index funds or ETFs. A simple portfolio of as little as two ETFs can give you a fully diversified portfolio of bonds by type (government, mortgage, corporate, domestic and international, etc.), credit quality (low to high), and maturity (short to long). These funds are typically held in tax free accounts like IRAs because the interest from these bonds is fully taxable. For assets held in taxable accounts, municipal bond funds are a great choice, providing either federal or state and federal tax-free interest to the tax-sensitive investor.