谢邀!
Good question. The differences among these can be confusing. I will explain them using examples. (Please pay more attention to "the other" and "another".)
"Other" is used to talk about a person or thing that is different from what was previously mentioned.Other than those few mistakes, your homework is perfect.One other thing I want to mention is that you look lovely today."The other" is "other" with a definite article (the). This indicates the number of other things is known or specified.My team didn't win. The other team won. (There were only 2 teams that could have won and the other team did.)The other three schools rejected me. (We know 3 schools rejected the person.)Another is "an" and "other" put together. It is "other" with an indefinite article (an) before it. This indicates the number of other things is unknown or unspecified.My team didn't win. Another team won. (There were several teams that could have won and some team other than mine won.)Another school rejected