If you appreciate something, you recognize its value. We can all appreciate how refreshing cold lemonade is on a sweltering summer day.
The oldest meaning of appreciate is "to know the worth of something": "The parolee appreciated his newfound freedom." A related sense is "to be grateful for something," but this usage can sound wordy. Instead of "I would appreciate your sending me more information," consider: "Please send me more information." The word can also be used (without a direct object) to mean "to rise in worth": "The vintage Corvette would appreciate in value over the years."