Thoughts on This Encryption Method, Please?
That looks like a (polyalphabetic) substitution cipher, with a less-straightforward method of entering the key. As already mentioned in the comments, close to the Vigenère cipher, which is rather well known. IIRC ciphers like that were used (and broken) in times before WWI (or WWII?), The Code Book by Simon Singh has material about historical ciphers, among other books, if you are interested.

As for not knowing the algorithm, it's usually taken as granted as the algorithm will be known (Kerckhoffs's principle), as it's hard to keep it hidden if the system is used widely enough and there is sufficient interest in it. Either the algorithm is widely available and can be bought, or it will be found out by old-fashioned spying, or in case of a single criminal using it, law-enforcement will be happy to read the software off your computer... (Since you need to have an unencrypted copy of the encryption program anyway.) 

As for the system, consider that using words as keys isn't that unpredictable, passwords are attacked with dictionaries daily, and the range of 26 letters out of 256 values is also quite small (if mostly lower-case letters.) Also, this isn't strictly true: 


  To decrypt the file, you had to reverse the process [...], using the original passwords in reverse order.


The algorithm has a property that makes the order of the keys irrelevant. Perhaps you can find it?