Answer:
The term "megabyte" originally meant 2^20 (1024 * 1024 = 1048576), and kilobyte meant 2^10 (1024). Then people who sold hard disks and other storage decided they could just count a megabyte as a 1,000,000 bytes, and make less storage sound like more, because that's how the prefix "mega" is usually used.
Since that's a pretty sound argument, two different values existed for kilobyte, megabyte, and gigabyte, until the newer set won out, and they decided to create new terms for 2^10, 2^20, and 2^30. They decided on kibibyte, mebibyte, and gibibyte. The "bi" syllable injected into each of the words is to allude to the word "binary," because they are all powers of 2, and thus the maximum values able to be represented by certain numbers of binary digits, or "bits."
The problem is that not everyone has accepted the new labels, so sometimes a megabyte means 1,000,000 bytes, and sometimes it means 1,048,576 bytes. If you're buying a hard drive or something similar, it will probably tell you, and it will probably be 1,000,000.
In conclusion, a mebibyte is always = 1,048,576 bytes, but a megabyte could be either that same value or 1,000,000, depending on the situation. Sometimes you just need to do the math yourself.