Kylo Ren's lightsaber is very confusing. It has a cross-guard, which was discussed in another question, but I don't think this was:
Why does Kylo Ren's lightsaber look different from a normal lightsaber? Kylo Ren's lightsaber looks like stuff is coming out of it. It looks like it's fire. Why?
TFA Visual Dictionary addresses this on page 26 "Kylo Ren: The Dark Warrior":
The blade looks this way because it is unstable plasma blade Matrix
This has to do with three facts also quoted on the same 2 pages:
Though Kylo's weapon evokes an ancient feel, its components are modern. Its crude appearance suggests construction by an inexperienced hand
The stressed crystal barely contains the power...
(this second fact also is relevant to the much-discussed cross hilt thing, but that's off topic)
Most importantly, the lightsaber uses cracked kyber crystal, which is the main cause of the ragged, unstable blade appearance.
Image for proof:
Kylo Ren's inexperience in lightsaber construction led to an unusual looking lightsaber.
From The Force Awakens Visual Dictionary:
In The Force Awakens we learn that Kylo Ren
betrayed Luke before his training was complete.
That is why his lightsaber, blades and hilt, seem very rough and makeshift. His blades appear unstable and even look as if they were on fire whereas most saber blades are smoother. His hilt looks jagged and makeshift while most other hilts are smooth and shiny.
In the question Why did Luke use a synthetic crystal in his lightsaber and what were the implications of this choice? we learn that even Luke's lightsaber ended up green because he didn't have access to a non-synthetic crystal.
It makes sense that leaving before completing your training would limit your access to certain materials you'd need to build a lightsaber. It would seem Kylo Ren may not have learned all there was to learn from Luke about making a lightsaber before he left.
Since the question doesn't specify in-universe, I'd like to add the out-of-universe answer to supplement the comprehensive in-universe answers so far.
Out of universe, the answer is that J.J. Abrams took the advice of Jony Ive, chief design officer at Apple, that a "more primitive"-looking lightsaber, which lacked precision, would look "more ominous". As The New Yorker reported in February of 2015,
...Ive once sat next to J. J. Abrams at a boozy dinner party in New York, and made what Abrams recalled as "very specific" suggestions about the design of lightsabres [sic]. Abrams told me that "Star Wars: The Force Awakens" would reflect those thoughts, but he wouldn’t say how. After the release of the film’s first trailer—which featured a fiery new lightsabre [sic], with a cross guard, and a resemblance to a burning crucifix—I asked Ive about his contribution. "It was just a conversation," he said, then explained that, although he’d said nothing about cross guards, he had made a case for unevenness: "I thought it would be interesting if it were less precise, and just a little bit more spitty." A redesigned weapon could be "more analog and more primitive, and I think, in that way, somehow more ominous."
I think we can see from the film that Kylo Ren's lightsaber is indeed "more spitty" than any other we've seen, as noted in the question's mention of "stuff coming out of it". It certainly does suit the character of its reckless owner very well, and in the history of the film's production, credit for that concept seems to go to Jony Ive.
I think Luke recognized Ben Solo's genuine interest in his grandfather and his force achievements. And foresaw the path it would ultimately take him.
So before introducing Ben to the lightsaber (Anakin's lightsaber) he stopped Ben's training and hid Anakin's lightsaber well away from Ben.
But as he is a determined young Force user seeking completion of training...he seeks out whatever parts and pieces necessary to make his own....even if the only power crystal available was a cracked synthetic crystal scavenged from somewhere...
This would explain why he loses his marbles and proclaims "That belongs to me!" When he first sees Anakin's lightsaber. Like a kid who always wanted it but wasn't allowed to have it. Obviously he's seen it before and truly believes he has a right to it.
It could be that his lightsabers blade reflects his internal conflict, his blade looks cracked and serrated, and because the crystals inside of the lightsaber and the beam itself are a reflection of ones connection with the force, it may be that when he has chosen either the dark side or the light side, that his blade will become smooth and true, but for now he is stuck with a unique awesome one!
