The 256 refers to the outcome of the SHA-256 algorithm always being 256 bits, so 256 elements of either a value 0 or 1. For readability you could write them down like '010100100010001001011101111..' but that does not mean the algorithm produces a 256 character string as you think.
So once these 256 bits are known you can represent those in many ways. One way is to write them down in hexadecimal where 8 bits are grouped into a byte and this byte is represented by a hex value. For instance:
Imagine the private key starts with these 8 bits:
11110110
The hex represenation of that would be:
f6
Since a private key consists of 256 bits you will get 32 (256/8) of these "letters (a-f) and numbers". Written down in human dreadable form in 64 characters.
The exact same key is often encoded in Wallet Import Format (WIF) as well, this is what you (probably) saw when generating an address on the bitaddress site. If you copy the WIF and paste it in the "Wallet Details" tab you can check out the full private key as well.
All in all: SHA-256 always produces a 256 bit result, this result can be encoded in many different ways but they all will refer to the same outcome.