"Another" is the same thing again. "Can I have another piece of pizza?" this indicates that they had a first, so now they want another (a second) "I got another cat." means that they have at least 1. Other is like saying, "but that" in some cases. "Can I have something other than pizza?" something brsides pizza. "I can't get any pets other than cats." Make sense? This was my sad attempt at explaining...

Another can, indeed, mean one more of the same thing, as in the pizza example given above. But it can be used as well to describe something a little bit different, but in the same category: " I don't like this chair, it is very uncomfortable. Do you have another one?" it is like taking "an" and "other" and making one word! "Other" by itself means something different, an alternative. " I did not like that restaurant, but I loved the other one we went to last week."