Even before the finale, Season 7 of Game of Thrones more or less confirmed that Jon Snow is the rightful heir to the Iron Throne, and is indeed the Ruler of the Seven Kingdoms. Sunday's season finale, however, goes one step further as it finally details Jon's true origin and birthright as Aegon (VII) Targaryen. That's right, we finally learn Jon's real name, although the showrunners take a few winding roads to get us there.
What this does is not just give us the Targaryen with the most powerful claim to the throne, but it also ties back the entire mythos to the ousting of the Mad King Aerys (II), and builds on not just the Targaryen succession, but also on Jon's legacy as the lone surviving male of the Targaryen dynasty -- which further punctuates how far ahead the television series is of George R.R. Martin's books. This reveal, arguably the series' biggest one to date, leaves us with a lot to analyze from both mediums!
In the finale, Samwell Tarly, Jon's best friend who sent the dragonglass information from the Citadel that drove him to partner with Daenerys Targaryen, arrives at Winterfell and converses with the all-knowing Bran Stark aka the Three-Eyed Raven. To put everything into context, we must remember that Aerys (II) sat on the throne, and was opposed by Robert Baratheon (Cersei's dead husband), the Starks, and the Lannisters. Their rage was driven by the thought that Aerys (II) was protecting his son and crown prince, Rhaegar (who was thought to have kidnapped Lyanna Stark, Robert's betrothed and Ned's sister) by killing Ned's father and brother.
This led to Jaime Lannister, Hand of the King, killing Aerys (II), allowing Robert to usurp the throne. The Targaryen family was now crumbling and Aerys' (II) queen, Rhaella, fled to later give birth to Viserys and Daenerys. However, Bran divulges that Rhaegar didn't kidnap Lyanna - they were actually in love, and Jon was the product of their romance.
The revelation of Jon's parentage stuns Sam, and Bran insists Sam find Jon to tell him the truth about not being Ned's bastard. As we saw last season, Ned lied about Jon being his bastard son to protect who we now know to be his nephew. Things get even more interesting from there because Bran adds that as Jon was born in the Tower of Joy in Dorne, and his father Rhaegar was married at the time to Elia Martell, Jon's real name should be Jon Sand (as Sand is the designated last-name to a bastard born in Dorne).
However, Sam has a bombshell of his own, which Gilly spotted in episode five this season. Recognizing Rhaegar's name, he recalls Gilly's findings in the Citadel that the prince's marriage to Elia was annulled and he urges Bran to use his powers to confirm this. Bran obliges, and he ends up discovering that this is true, and even sees the wedding of his aunt to Rhaegar, confirming that not only is Jon his cousin, but according to Lyanna's words to Ned, Jon's true name is actually Aegon Targaryen! These revelations are ironically juxtaposed to Jon consummating his relationship with Daenerys - his biological aunt.