Rachmaninov's Rhapsody on a Theme by Paganini Op 43 in A minor, is romanticism at it's best. It is a work for solo piano and orchestra closely resembling a piano concerto. The composer wrote it whilst in Switzerland in 1834, as a keen interpreter of his own works he played it at the premiere at the Lyric Opera House in Baltimore with the Philadelphia Orchestra with whom he also made the first recording.
A variation is when a catchy tune can be pinned down and strung along by repetition and embellishment. Variations are usually lighter in intellectual substance than a composers greatest works and could be said to be more decoration than architecture. However some composers reach through the superficiality of the structure to find great heights, as is the case here.
Paganini's Caprice No 24 for solo violin has been used by many composers as a theme to be varied for a whole variety reasons. Firstly it is in the clean key of A minor, the theme begins and ends in the same key and in the middle it uses a cycle of 5ths. The cycle of 5ths was much used by Baroque composers as it enabled them to repeat a simple theme whilst changing key each time, thereby giving Baroque music it's rich harmonic sound. Listen to Hilary Hahn playing an excerpt of the original on YouTube.
Rachmaninov wrote the Rhapsody when his popularity was waning in the eyes of audiences and critics. Since leaving Russia for the west he had written little and at the time his 4th Piano Concerto was considered a flop. His romantic way of writing was out of fashion as music was becoming much more modern with composers such a Shostakovich and Prokofiev. However, even those who don't particularly like Rachmaninov's music marvel at this work.
Throughout the work there is one recurring theme which is the Dies Irae from the Requiem Mass, normally a symbol of doom and judgement, here it is used to give the work great stability and strength. It can be heard through the variations.
The recording I recommend is played by the Macedonian pianist Simon Trpceski with the Royal Liverpool Orchestra conducted by Russian Vasily Petrenko, their principal conductor. This is an excellent recording and also contains Rachmaninov's piano concertos now 1 & 4, both also great works.
"Trpceski is alive to the sheer fun to be had and his impetuosity is neatly matched by Petrenko's Liverpool Players" The Arts Desk 2011
Here is the brilliant Stephen Hough talking about and playing the Rhapsody on a Theme by Paganini on YouTube