1 An Unearthly Child - This is the first episode of Doctor Who and was broadcast on November 23rd 1963. It beautifully sets up the long running series and introduces both the now famous Doctor (as played by William Hartnell) and his granddaughter Susan.
2 The Daleks - This is the second Doctor Who story and introduces the Doctor's most iconic enemies.
3 The Tomb of the Cybermen- This is one of Patrick Troughton's Doctor's greatest stories and helped to inspire Matt Smith when he first took on the role.
3 Carnival of Monsters - This is an atypical serial for Jon Pertwee's Doctor who finds himself trapped along with his companion in a zoo of miniaturized creatures. This story sais a lot about important social issues and works well on the front of entertainment.
4 The Three Doctors - "The Three Doctors" brings the original three incarnations of our hero together for one adventure. While the first Doctor might not actually be physically present and some of the enemies may be incredibly cheesy this story is definitely worth a viewing.
5 Pyramids of Mars - Tom Baker is for many the definitive Doctor. In this story we see him battling robotic mummies and an alien death god alongside his beloved companion Sarah Jane Smith.
6 The Caves of Androzani - Peter Davison's Doctor seems to have been the original youthful and charming incarnation. This serial is his final story and is considered by many to be one of the greatest stories of the entire series. We see the Doctor make a sacrifice most anyone would struggle with and Davison's performance comes across as extremely genuine.
7 Vengeance on Varos- The sixth Doctor (played by Colin Baker) is widely considered the least liked incarnation. While some of his stories may be lacking in many ways this one is worthy of much praise. "Vengeance on Varos" is an arguably political examination of cultural views on violence and the nature of good and evil. You might not necessarily find it an easy watch but it has a lot to say and is a really strong story.
7 Dragonfire - The seventh Doctor (played by Sylvest McCoy, of Hobbit fame) is in my opinion one of the most underappreciated of all the Time Lord's incarnation. His stories were not always the best but he had some really great ones and Ace was a fantastic companion. McCoy's portrayal was originally almost entirely comical but later his Doctor began to manifest a noticeable dark side which has remained ever since. "Dragonfire" is basically space Dracula. Also introduces the character of Ace and has a really special effect towards the end.
8 Rose- This was the first story of the revived series and it not only introduces Chris Eccleston's Doctor but also the iconic companion Rose Tyler. This story also marks the return of the autons who were a popular enemy of the classic series. This episode is very good if a little cheesy at parts and is probably one of the single most important stories in the series as a whole.
9 The Empty Child/The Doctor Dances - This two parter by future showrunner Steven Moffat is a creepy and well told story centering around the London Blitz. It asks an important question about the Doctor and introduces the character of Captain Jack Harkness.
10 Bad Wolf/The Parting of Ways - This two parter concludes the first series/season of the revived Doctor Who as well as Christ Eccleston's run of the Doctor. It brings to a head a mystery which was building the entire season and features an awesome confrontation between the Doctor and the daleks. It concludes with the Doctor once again making a heroic sacrifice and regenerating in the TARDIS
11 Army of Ghosts/ Doomsday - This two parters marks the end of the second season of the new Doctor Who. It is also the end of Rose Tyler's time as a regular companion.
12 The End of Time parts 1 and 2 - This is David Tennant's regeneration story. It's a wonderful and epic two parter which serves as a close to perfect goodbye to a true great.
13 The Eleventh Hour - This is Matt Smith's first episode and introduces both Amy Pond and Rory Williams as well as a very important crack and a morbid prophecy. It's full of laugh and excitement and has a lot packed into it.
14 Vincent and The Doctor - This is about Vincent Van Gogh and his emotional troubles. It is a beautiful story and is very human.
15 The Pandorica Opens/ The Big Bang - This two parter ends season five of new Doctor Who and is both very fun and exciting as well as dark and emotional at times. It ends the season on a high and is a really fun watch.
16 The Doctor's Wife - This is the story of the Doctor's greatest love. It is also a creepy and claustrophobic tale written by the extremely talented Neil Gaiman.
17 The Day of the Doctor - The 50th anniversary of the series was rightfully marked by this action packed special. This story stars three incarnations of the Doctor (David Tennant's, Matt Smith and John Hurt's mysterious war Doctor) and features all the other fully canonical versions. The zygons who have been beloved ever since their original appearance in a fourth Doctor story return and their are lots of great nods to the past. This story concludes by sending the Doctor in a new and interesting direction.