Entropy is a measurement of the appearance of "randomness" of bytes. For example, if you were to take the entropy of the text content of this post, and then take the entropy of a SHA-512 hash or RSA ciphertext, you will see a dramatic increase in the hash and RSA ciphertext over the plaintext content of this post. There are known entropy levels for plaintext English, for instance.
How it can be used in reverse engineering
One useful function of entropy is to determine areas of ciphertext or other encryption outputs in a program. A concrete example would be to take a "packed" or compressed malware sample, and compare the entropy levels to a non-packed or non-compressed program. The packed malware sample will have entropy levels that are much higher. One useful application of this would be to simply write a script which batch-scans files and takes entropy readings, in order to identify files that have large areas of packed/compressed/encrypted data. These files will have higher entropy readings. A malware analyst will sometimes take an entropy reading of a file in an initial cursory analysis to determine if the file is packed or not.
Switching gears, if you wanted to pack or compress data in your own program, you could write your packing code, then compare the entropy of the output data with the entropy of the uncompressed data to see how much "randomness" you've introduced.