Other: it's mainly used to differentiate people or things from something already talked about. Example: "I don't want that cookie. I want the other cookie."
Another: Think of it as an addition to something, like add one more of something already talked about. Example: "I want another cookie."
The others: Think of this one as, "those that weren't already mentioned in the group." You're trying to differentiate from things, people or places that have already been mentioned. For example: The others were late to the field trip, so they had to stay at the school." What makes this different from other is you're talking about more than one thing, person or place as you would with other. Others=more than one thing, Other=one thing