Other means a person or thing that is different from one that you have already mentioned. For example: "I don't want this car, I want the other one." Here there are two cars available. This person was offered a specific car but preferred a different one. If you were to use "another" car here you would be asking for two cars.
The second meaning of other is an additional person or thing. So you might say, "One reason I like honey is because it is sweet. The other reason I like honey is because it is healthy." Here if you use "another" instead of "the other" you change the meaning from a specific reason to a non-specific reason.
Another means "in addition to one or more of the same kind." In this case you would say something like, "I want another chance." In this case you have had one chance and you want a second chance in addition to the first one.
Another may also mean "some other." An example of this use would be, "We can talk at another time."
The last meaning of another is "similar or equal to a particular person or thing."
An example of this is, "When viewed from another angle, the painting looks entirely different." Here we are comparing similar things, angles.