As the name implies, a Theme is a musical form which is repeated over and over but has changed somewhat with each repetition. The changes may evolve around the melody, harmony, rhythm, or tones of the composition. Often the theme's bass line is repeated while 'variations' of the melody or harmony are played above it. Film and movie themes are a prime example of the Theme Song with variations.
Many renowned Classical symphonies are written in the style of the Theme Song. J.S. Bach wrote 'Goldberg Variations' to help insomniacs fall asleep. It is said that Bach was greatly rewarded for the composition by those finding it of personal value. Many other composers then followed Bach writing theme variations of their own.
While composers like Haydn included small variations in their composing, others such as Mozart wrote exceptional variations for various instruments or orchestral accompaniments. Brahms on the other hand, wrote several variations on already existing themes from the aforementioned composers and others such as Handel.
Toward the end of the 19th century Elgar composed his famous 'Enigma Variations,' a form which continued into the 20th century as a popular composing style. One of the best known themes from the period is Rachmaninov's 'Rhapsody on a Theme of Paganini.'
Both renowned and upcoming composers seem to love the Theme & Variations style of writing so its popularity and usage continues up to today.
'Goldberg Variations' (BWV 988)
Composed by Johann Sebastian Bach