the general idea is to create an encryption algorithm that can be decrypted 2 different ways. one way (the front door), uses an encryption key (like a password), that the encryptor chooses. the other way (the back door), uses an encryption key that the algorithm designer chooses (or at least knows). you can think of the back door key as a master key that is built in to the algorithm, but not obvious to someone analyzing the algorithm. as such, anyone using the algorithm can pick their own secret key, then only people who know that key or the master key can decrypt the message.
to give a slightly more concrete, yet approachable example, you must understand the difference between symmetrical and asymmetrical encryption. symmetrical encryption uses the same key to encrypt and decrypt a message. asymmetrical encryption uses a pair of "matching" keys, one to encrypt, and the other to decrypt. with asymmetrical keys, you can tell everyone the encryption key, and they can encrypt messages that only you can decrypt. however, symmetrical encryption tends to be faster, so in practice, people usually use asymmetrical keys to encrypt a symmetrical key, then use the symmetrical key to encrypt the actual message. using this strategy, you could simply encrypt every message with a random symmetrical key, then attach a header to the message with a copy of the symmetrical key encrypted using your asymmetric encryption key and a copy of the symmetrical key encrypted using the "backdoor" asymmetric encryption key.
based on the above, i think it is fair to call a "back door" a hole rather than a weakness. you might argue that some of the cryptographic weaknesses promoted by the nsa could be called a "back door". but since it takes a billion dollar budget to leverage the weakness effectively, it is more akin to export controls on strong encryption rather than a back door. in either case, it is not typically effective to "hide" the back door. generally the academic community at least suspects the existence of a back door long before an encryption algorithm becomes a popular standard. it is merely the key to the back door that remains secret.