This is taken from an article I wrote (and was paid for) at Associated Content. These are in no particular order, also not necessarily MY favorite episodes but five solid episodes that sort of define the series and would be a good starting off point for say, someone who is unfamiliar with the series. Also, not they they are good enough to be included, but this is from early 2010 and does not include the DVD movies or Season 6.
Season 1, Episode 3: Synopsis: Fry is kicked out of the Planet Express office and moves in with Bender. After an exhaustive apartment search the duo fall into a lavish pad that turns out to have satellite reception that is interfered by Bender's antenna, thus requiring a sacrifice by the robot to continue living with his friend. In the end they learn a lesson about friendship and ultimately they move in together back in Bender's old apartment with fry living in the closet and Bender complete again. Review: This Episode really kicked off the series and establishes the newly formed, yet strong bond between Bender and Fry. It also helps establish some of the other characters' personality such as Leela's nosiness, Amy's clumsiness, Zoidberg's disparity and it introduces us to Hermes as the Rastafarian office Bureaucrat as well as his disdain for Zoidberg. I especially liked how even in the future they still have the same foibles and upsides to having a roommate, all montaged with the "Odd Couple" theme.
Season 3, Episode 7: Synopsis: The Earth is attacked by evil flying brains and it is revealed that Leela's pet, Nibbler is actually a member of an adorable race sworn to protect the universe from the cerebral menace and that Fry seemingly has a much higher purpose than being a lowly delivery boy. Review: This is really the 1st episode in the series to address the main underlying story of the series about Fry's purpose in the future and his role as the only hope for the survival of the universe as well as exploring the Niblonians more. I think it is great because it showcases the amount of thought that David Cohen and Matt Groening put into the series as a whole from the beginning and although the 3rd season is a little late it starts a great story arc that has been the backbone some wonderful episodes.
Season 3, Episode 10: Synopsis: An unlucky fry goes to his parent's old basement in the ruins of Old New York to recover his lucky seven-leaf clover that in his child hood allowed him to best his brother, Yancy at basketball and break dancing. During Fry's search he is led to believe his brother stole the clover for himself and led a fabulous life only to discover some shocking revelations about the past in his search. Review: This episode is one of my favorites for the great story. I always enjoy the episodes that expand on Fry's world in the time surrounding his entrapment in the cryo chamber and as a person with siblings the introduction of Yancy made for some funny segments during their childhood that were also very relatable. It is also is a great episode because the ending is very well done and bittersweet. It showcases the ability of the writing staff as being able to produce much more than nerdy jokes and obscure sci-fi references.
Season 4, Episode 1: Synopsis: During an accident caused by Fry's misuse of a microwave during a Supernova the crew is transported back in time to 1947 where they reveal what really happened that year in New Mexico. Review: This list would not be complete without "Roswell That Ends Well." In addition to being an Emmy winning episode for Outstanding Animated Program (Programming Less Than One Hour), this episode has a wonderful and fun story and is beautifully animated. The time travel plot device allows for great jokes and references from Star Trek to Gomer Pile and the revelation of who Fry's true grandfather is given as the reason he is so effective against the giant brains in a later episode. The side story involving the Professor and Leela seems kind of forced, but Zoidberg's story as the alien that was captured at Roswell is classic and makes for great filler between the scenes involving Fry and his relatives.
Season 4, Episode 12: Synopsis: After learning that his favorite television program from the past, Star Trek was banned during his time frozen; Fry, Leonard Nimoy's Head and the Planet Express Crew search for the lost episodes of the forbidden program only to wind up on a planet ruled by an energy being that has been keeping the original cast of the Star Trek series as collectibles. Review: This was a great episode because it brought together most of the surviving cast members from the show that has provided Futurama with so many of its jokes and plot devices. The banter between the old Trek Crew and the Planet Express is priceless and I loved the idea of having an antagonist that could have been straight out of the old series turn out to just be a Trekkie living in his mom's basement.