Who is the Last Jedi?
This is a much more complicated question than it initially appears, mainly because the definition of "Jedi" itself seems to be a bit questionable. Who gets to determine who is a Jedi and who isn't?
One thing that is clear is that not every single Force-user is a Jedi or Sith. During the Clone Wars there were only 10,000 Jedi Knights and Masters, out of a galaxy with a population of quadrillions. Even with only a tiny decimal of a single percentage point being able to use the Force, that's still millions upon millions of potential Force-users at large in the galaxy at any one time. Therefore the suggestion from the original trilogy that Yoda, Luke and Obi-Wan were the only light side Force-users of any significance in the galaxy, and the Emperor and Vader were the only dark side Force users of any significance in the galaxy, was already highly doubtful. What was more likely meant was that Yoda, Luke and Obi-Wan were the last Jedi, or the last of that tradition, and Vader and the Emperor were the last Sith. The latter was given more credence in the prequel movies which confirmed that there are only ever two Sith around at one time.
When it comes to the Jedi, there was more of a formal hierarchy in place. The Jedi Council was in charge of promoting people from the rank of Padawan to Jedi Knight, and from Jedi Knight to Jedi Master (and a seat on the council). The Council members therefore had the power to name Jedi. Shortly after the Clone Wars began Obi-Wan Kenobi was promoted to the rank of Jedi Master: more specifically, after the events of Attack of the Clones but by the opening episodes of The Clone Wars animated series, during which time he is already on the Council. Yoda, of course, was already a Master. The entire order of Jedi Knights and Masters was wiped out by Order 66 during the events of Revenge of the Sith, bar only Obi-Wan and Yoda, so as of that time they constituted the entire Jedi Order and the Council.
In Return of the Jedi, after Obi-Wan's death, Yoda is the sole surviving Jedi Master and member of the Council. He officially names Luke Skywalker as a Jedi Knight (although Luke had already been calling himself that to impress Jabba earlier in the movie) mere moments before his death, and that seems as official as things need to be.
What happens after that is less clear, but it appears that Luke tried to train a new generation of Jedi apprentices and these were all murdered by Kylo Ren and his Knights of Ren. As far as we know right now, no other Jedi Knights or Masters were named during this time period by Luke. Based on supplementary material and interviews, it appears that Leia, although Force-sensitive, chose not to explore her Force abilities in favour of her political career and thus never entered the Jedi tradition.
This seems pretty straightforward then: as of Star Wars Episode VIII: The Last Jedi, Luke is the last Jedi. Whether or not he trains Rey and eventually names her a Jedi Knight will, presumably, be an important story point in both The Last Jedi and Episode IX.
(Also, yes, the Force Awakens title craw says outright that Luke is the last Jedi. But that would have been a much shorter article.)
Of course, there are some complications with this.
Ezra Bridger, Kanan Jarrus and Ahsoka Tano during the events of Star Wars: Rebels.
Lucasfilm and Disney are very, very clear that the animated spin-off series Star Wars: Rebels is 100% canon even in their new continuity, as is its predecessor series, The Clone Wars. Up to a few months ago, people would have taken that with a pinch of salt, but the movie Rogue One has pretty much enshrined the show in the movie continuity. The starship Ghost shows up in the Rebel Fleet in that movie, the psychotic droid Chopper shows up in the Rebel Base, Hera Syndulla (now a general) is name-checked and the Hammerhead corvettes stolen in Rebels show up and play a key role in destroying two Star Destroyers during the Battle of Scarif.
The problem with this is that Star Wars: Rebels features no less than three Jedi - or light side Force users - in central, key roles and working alongside the nascent Rebel Alliance. As Rebels is aimed a younger viewers, the general assumption is that the show is not going to brutally murder its entire cast as the show draws to a close, which leaves the fate of those characters in doubt and how they relate to the title "Last Jedi".
The most established of the three characters is Ahsoka Tano. She was Anakin Skywalker's padawan apprentice during the Clone Wars and became a skilled and brave member of the Jedi Order. However, some months before the end of the war she was framed for a crime she did not commit. She proved her innocence, but was so disgusted with the Jedi Order not believing her innocence (apart from Anakin) that she quit the order and went into self-imposed exile far across the galaxy, completely missing the end of the war, Order 66 and the rise of the Empire. Crucially, Ahsoka was never made a Jedi Knight, so was not officially considered part of the order. Fifteen years later Ahsoka returns to prominence during the formation of the Rebel Alliance, having become a far more formidable Force-wielder. She injures Darth Vader in single combat (after learning he is really Anakin) and escapes certain death at his hands, but was last seen trapped in a Sith temple on a remote planet. Her fate remains to be explored in Rebels.
The next most-established character is Kanan Jarrus. A padawan during the Clone Wars, Kanan watched his mentor and Jedi Master killed in front of him during Order 66. He barely survived and fled into deep exile and cover. As an apprentice with only light experience, he was forced to improvise his own training. Many years later he joined the Rebel Alliance. During a mission to the planet Lothal he met a young man, Ezra Bridger, who was strong in the Force. Despite misgivings, Kanan started training him as a Jedi, taking him as an effective padawan (despite Kanan himself never being given the rank of Jedi Knight). Kanan's attempts to train Ezra were complicated when he was blinded in a lightsabre duel, leaving Ezra to take more training onto himself...rather dangerously, after Ezra came into possession of a Sith holocron containing forbidden knowledge. This storyline remains in play on Rebels.
Given the events of the original Star Wars trilogy and the newly-revealed title, it seems that what fans had been assuming about Rebels is confirmed: Kanan, Ahsoka and Ezra don't make it to the attention of Luke during the original trilogy and they never become Jedi Knights (or, if they do, they're dead by the time The Last Jedi rolls around). Whether they live, die or survive but are cut off from the Force remains to be seen. However, the show itself does give us a possible explanation. In Season 2 of Rebels it is revealed that the Emperor is not only hunting down former Jedi and apprentices, but also children who show Force abilities. In Season 2 Ezra and Kanan rescue two of these younglings and get them to safety. One way of evading the issue is that either or both of our heroes have to take these younglings into a remote part of the galaxy to help them train. If this is the case, they may have an excuse to sit out the sequel trilogy and may even show up afterwards to help Luke found the nascent new Jedi Order. Another possibility is that Ezra himself turns to the Dark Side and has to be taken down, but this would again be quite dark for a kid-oriented show.
Time will tell where the story goes, but this title of the new movie certainly does not bode too well for our Rebels heroes.