[susan53]

I'd disagree slightly with Tania's examples. "The other" isn't necessarily only used with two things/people. There can be any number of them but "the other" specifies the last remaining one to be discussed :

I have three brothers : One lives in Italy, one in Germany and the other in the States.
With this example I could also say ... another in Germany and another in the States. With more than two I have the choice of whether to present them all as indefinite or the last as definite.

In Tania's examples of the handbag and example , the indefinite form is used because I am not asking for a specific handbag/example but any handbag/example the person wants to give me.

So basically, the choice between another /the other is exactly the same as the choice between the indefinite and definite articles with any other noun phrase. If I'm talking about something specific, then "the" :
I don't like these pens. Give me the red pen / the other pen.
I don't understand this rule. Give me a clearer example /an easier example / another example.

The fact that the indefinite article "an" is written together with "other" to form one word, doesn't change the fact that really it's just an "ordinary" indefinite article + noun phrase construction - though the noun is often elided (omitted) because it's understood in the context, as in :

I have three brothers : one lives in Italy, another (brother) lives in Germany, and the other (brother) lives in the States.