"Uhr" is a general term for all devices that show time. In English, these are called "clocks" and in some instances "watches". "Armbanduhr" is a more specific term for a "Uhr" you wear on a "Band" around your "Arm". In English, that's a "watch". But while in German one could say "Du hast eine schöne Uhr an deinem Arm", in English you can't say "You got a nice clock on your arm". So both German and English have two terms, but the usage differs.

For compound nouns in German, the gender is the same as the 'regular' noun it ends in. Die Uhr, so you know that Armbanduhr is the same gender, since it ends with Uhr. =Die Armbanduhr it's just like Der Ring and Der Ohrring

Dirk already answered the question, just one more detail: it's also *die* Armbanduhr.

DIE Armbanduhr !!! Aber DAS Armband.... und DER Arm. Tja, 德说不容易