in short a private key is simply a number and no matter how you encode it, in the end it is still that one number. it could be base-16, base-64, base-58, or even bech-32.

for each private key there is one public key that can be encoded in two ways: compressed and uncompressed.
in bitcoin we can create many different script types in a transaction output script with that public key. some of these have an address associated with them. to make it easy for wallets to quickly figure out which public key to use and which address to import a different byte (or bytes) is added to that number (the private key number) before encoding it with base-58 so that the result looks different but is still representing the same number. if it is starting with "5" it is for uncompressed public keys and if it starts with "K" or "L" it is for compressed public keys.

bitaddress.org supports 4 private key formats:
base-16 or hexadecimal which is the encoding of the number (without any addition) in base-16
base-64 same as base-16 but with a different character set
base-58 which is special encoding used in bitcoin with a checksum
minikeys which is a special short format of private keys used in some collectibles (physical bitcoins)

you should use compressed base-58 as it is supported by all wallets so it can be imported in all of them.