Simply put, realized profits are gains that have been converted into cash. In order words, for you to realize profits from an investment you've made, you must receive cash and not simply witness the market price of your asset increase without selling. For example, if you owned 1,000 common shares of XYZ Corporation, and the firm issued a cash dividend of $0.50 per share, you would realize a profit of $500 from your investment. This is a realized profit because you have received the actual cash, which cannot be lost due to changes in the marketplace.
Similarly, let's say you purchased your 1,000 XYZ shares at $10 per share, for a total investment of $10,000. If XYZ Corp. were presently trading on the market for $15 per share and you sold all of your 1,000 shares on the open market at $15, you would realize a gain of $5,000 on your investment ($15,000 - $10,000).
Now, suppose that XYZ Corp.'s shares were trading at $15, but you believed they were fairly valued at $20 per share, and therefore, you were not willing to sell at $15. Because you would still be holding on to all of your 1,000 shares, you would have an unrealized, or "paper", profit of $5,000. Of course, if you have not closed out of your position and realized your gain, you could still lose some, or all, of your profits - and your principal as well.
On the other hand, because you have not realized your profit, you are not required to claim the gain as income; thus, by holding your shares instead of selling, you can potentially defer taxable income for a year (or many). Of course, the reverse is true for losses - realized losses can usually be claimed by investors as capital losses, offsetting other capital gains, while paper losses can not.
To learn more about profit realization and its implications for investors, check out Selling Losing Securities For A Tax Advantage, A Long-Term Mindset Meets Dreaded Capital-Gains Tax and Tax Tips For The Individual Investor.