Do I need to know the "mode of operation" to decrypt a message encrypted with a block cipher? If I have received an AES encrypted message, and if I do know the key with which it was encrypted, do I also need to know the mode of operation with which it was encrypted in order to decrypt it? My understanding is that this is indeed the case.

If so, how is the mode of operation transmitted along with the encrypted file?
Yes, of course you need to know the mode of operation in order to decrypt.

On the other hand, the mode of operation isn't usually explicitly transmitted in the ciphertext.  Using the same key with different modes of operation may cause unexpected weaknesses.  For example, consider one message encrypted with CFB mode with IV=2 (which is perfectly secure) and another message encrypted with CTR mode with IV=0 (also secure).  If we give the attacker both ciphertexts, he knows a relationship between the first block of the CFB mode plaintext and the third block of the CTR mode plaintext; this is a leakage that does not happen with either mode individually.

Because of this, we restrict a specific key to one mode; that is, both sides agree that this specific key will be used with that specific mode.  If there are multiple modes possible, the two sides will agree to the mode at that time.

Now, there are some cases where a key is used only once; one such case happens we're using public keys to encrypt the file.  In this case, the encryptor will pick a random AES key, public key encrypt that AES key (using the receiver's public key), and then use the random AES key to encrypt the actual message.  In that case, you can encrypt the mode (and other details, such as whether we're using AES and not 3DES) along with the random key.