How can you find out why a relative was awarded a medal?

This would be listed in their discharge papers. If you don't have that you can PROBABLY get a copy by going to the National Archives and Records Administration website http://www.archives.gov/research_room/vetrecs/ and requesting it. I say probably because many records were lost in a fire in 1973 but they can usually find copies of discharge papers from other sources.

However, in the case of the Bronze Star, it may not be listed on the discharge papers. In 1947 Congress authorized all WWII recipients of the combat infantry badge and combat medic badge to receive the Bronze Star. Many vets weren't even aware they were entitled to the Bronze Star.

Other than the Medal of Honor and Distinguished Service Cross, there are no centralized registries of Military decorations.

Or you could just ask them

Some nations DO have central registries of military medals and awards, going back, in some cases, 175 years. In the British and Commonwealth nations, the awarding of honours and medals is recorded in the Offcial newspaper of Parliament, where all new laws and regulations are also published.

In Great Britain that publication is the London Gazette, in which every military award is listed after it has been presented to the recipient.

In France, the lists are kept by the Marshall of the Lists, in the main armoury in Paris.

In Canada the lists are kept by Veteran's Affairs Canada, and are available on-line thru their website. EVERY man who served Canada in World War One ( 1914 to 1918 ) has his records available to the public thru the Canadian Archives website, all 665,000 of them.

The information above is incorrect. A DD-214 (Discharge Certificate) should list all awards and decorations a servicemember received, but would NOT say what it was awarded for. A certificate awarded with the medal would have a "narrative" listing what the medal was awarded for. A copy of the Certificate would be in the Official Personnel Record.