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Base 256 is referring to the 256 values of a byte.
ASCII represents a single character in seven bits. For obvious reasons, each character takes up one byte with the most significant bit unused.
If you slap a random series of bits together and call it an ascii string then it will look pretty funky. A fraction of the available 256 values are printable characters.
This is where base 64 comes in. When you have arbitrary binary data, picture crypto salt etc, you might not want null bytes and unprintable characters and such in your output (Such as the many common cases in which such sequences are not allowable). It is a way of encoding this data such that you can send it in text-only mediums due to the fact that it results in a narrow selection of printable characters. Take email for example, specifically the SMTP protocol...
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MD5(sig) = bcef75433db02e9ad9bf81d6f7c5c270
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