Routine is the only one that sticks in your mind. A routine (procedure, function, method, operation) is a part of a program. (NB: there are other different parts to programs, too: threads, objects, queues, etc.) There is no necessary one-to-one mapping between a routine and an algorithm. If you wish to talk about something a program does, you can speak specifically about its functionality. Like the spell-check function in a word processor or an email client.

You can also have many different algorithms that solve the same problem, e.g., sorting. In the end you want your numbers sorted, which algorithm you choose to implement in your program will have an impact on things like speed, efficiancy, etc.

Ceci n'est pas un seing.