I'd like to expand Bob's explanation of verse and chorus by putting it into a historical context. In doing so, I'll probably become pedantic and long-winded (two of my less endearing traits), but here goes…
Traditional song ballads tell stories. Their structure is strophic, meaning that they repeat the same music with changing lyrics. One strophic form is the verse–refrain, exemplified by the traditional ballad "Yankee Doodle". In this case, the first verse begins "Yankee Doodle went to town" and the refrain follows "Yankee Doodle keep it up." The second verse begins "Father and I went down to camp" followed by the refrain. The third verse begins "And then we saw a thousand men" followed by the refrain. There are many other verse–refrain strophes to follow. In each case, the verse tells the ongoing story and the chorus sums things up.
A popular song example that most folks are familiar with is "The Band Played On," the 1895 hit by Palmer and Ward. In this case, in each of three strophes there is a story-telling verse followed by the refrain (or chorus): "Casey would waltz with the strawberry blond and the band played on…" A similar structure can be found in the 1892 hit "After the Ball."
Things changed in the 1920s with the rise of true musical shows. In contrast to reviews, with many separate "acts" performing their own material, music shows were organized around a story with songs specifically written to further the plot. In this case, the verse was most often not sung in rhythm and were written to set up the context for the refrain. Often the refrain would be repeated (perhaps by a different singer or with dance). While the verse had no specific structure, refrains were most often 32-bars long with either an AABA or ABAC structure. For examples, look at The New Real Book which usually includes verses with vocal songs.
As songs were adapted by jazz musicians for instrumental arrangements, the verses were generally omitted and the 32-bar refrain was repeated many times. Each refrain is referred to as a "chorus" and that is the terminology we use today.