Other is an adjective meaning 'different' or 'the second of two items'.  Another expresses quantity.  It means 'additional' or 'one more'. Example: I found one cat, I hope I can find the other cat soon (This implies there are are only 2 cats in total ) I found one cat, I hope I can find another cat soon. ( We don't know how many cats there are in total, there could be many cats)

Let see... First, “other” is plural and “another” is singular. It’s like a combination of the words “an” (which is singular) and “other.” Examples: “Let’s look at other options.” The noun being referenced (options) is plural. “I want another dog.” The noun being referenced (dog) is singular. Second, “another” can be used to show the addition of new subject. (That’s a weird way to put it). Examples: In the sentence “I want one more drink,” you could change it to “I want another drink,” because it is adding onto what is previously there. It is sort of like “I want an additional _________” for example. I hope this makes sense!