Who is the most popular poet of all times? What were some of their popular poems?
Certainly this is a question that is open to opinion and could be argued in several different directions. If you are writing a paper on this topic, I encourage you to define the term "popular" or, essentially outline the parameters for which you are measuring popularity.
I'm going to argue that William Shakespeare could be considered the most popular poet of all time. Certainly, he has been considered the greatest writer in the English language, and though he is most well known for over three dozen plays, he wrote poetry as well. He is probably most well known for his 154 sonnets. These sonnets are well-known, oft-quoted (in speeches, film, literature), and in many ways foundational for those who study, write, and write about poetry. In addition to sonnets, Shakespeare is also known for a couple volumes of narrative poetry, including the erotically themed Venus and Adonis and The Rape of Lucrece.
I think one of the strongest arguments for Shakespeare as the "most popular" poet of all time comes in the range and scope of his influence, during his time and through today. His work has not only impacted poetry and theater, but novelists as well. Finally, Shakespeare has highly influenced, and essentially changed, the English language. He is credited with helping to standardize several nuances in grammar, punctuation, and spelling. Further, several of phrases originally coined by Shakespeare have made their way into our everyday speech, as common idioms and figures of speech.
Without disagreeing with Shakespeare as an answer, one could argue that Edgar Guest is the most "popular" poet -- syndicated in dozens of newspapers for 3 score years. Considered doggerel perhaps by anthologists, but read daily by millions of newspaper readers in the 40's, 50's and 60's, Guest, like Ogden Nash, was "popular" in modern times. His poems have since been gathered into volumes.