The Spellplague was the third of the three great disasters to befall the Forgotten Realms, which served on a meta-level to justify the switch-up in mechanics and lore from one edition to the next. In this, it was the direct sequel to the Time of Troubles.
It is also, without a doubt, the most absolutely skubtastic aspect of Dungeons & Dragons 4th Edition, because of its massive - and unnecessary, in light of the more respectful treatments that 4e gave to Eberron and Dark Sun - adjustment to continents, metaphysics and characters. Whilst Faerun is infamous for having a particularly autistic fandom, the Spellplague didn't help by having a story that relied on way too many people either doing stupid shit or having more power than they should have just because the plot needed them to, with even fans who liked the basic idea agreeing that it could have benefited from a more competent backstory.
In a nutshell, as part of a long-running plan on his part, Cyric finally managed to arrange for the assassination of Mystra, one of his oldest enemies. Naturally, killing the Goddess of Magic did bad things to "The Weave", Toril's background magical fabric-thingy - it didn't help that Shar immediately tried to stick her hooks in it and claim it for herself. The result was that magic went completely haywire, causing magical spells and items across Toril to explode in surges of eerie blue flame that acted as magical radiation, complete with inducing bizarre physical and mystical mutations in anybody who was touched by the stuff and survived. Lots of people died, lots of shit got blown up real good.
The Spellplague was ultimately cleared up when The Sundering occurred, pulling Abeir and Toril apart again and moving things on to 5th edition.
The effects of the Spellplague, and in particular the Spellscarred theme, are technically a kind of Wild Magic.
Seriously, what happened?
Okay, for real, the problem with elaborating on what the Spellplague did is twofold. Firstly, not a lot of detail is actually gone into in the 4e Campaign Guide for the Forgotten Realms, which is pretty vague on the details for the most part - you have to read a lot of the tie-in novels and even some Dragon Magazine articles for some of the skubbier details, and in fact it can be hard to immediately know what's changed at a glance unless you have the 3e and 4e campaign guides side-by-side for comparison or are a Realmslore autist. Secondly, the actual changes to the 4e version of the Realms are only partially due to the apocalyptic wave of Wild Magic that was the Spellplague; the fact that time has skipped forward 100 years from 3e has also left its mark on the world.
Still, 1d4chan will try to provide an easy summary of shit that changed in between 3rd and 4th edition.
How It Happened
Cyric murdered Mystra and then Shar tried to take control of the Weave. That's the fundamental core of why the Spellplague happened. People complain about this, pointing out that two versions of Mystra had died and been reborn before this one, but that's also ignoring that both deaths also left their mark on magic as a whole. The death of Mystra #1 caused all magic to rapidly weaken until it almost vanished entirely, only returning after she reincarnated herself - whilst officially she banned 11th level and higher spells, one could read it that the Weave was so damaged by her death that such powerful magic simply became impossible to use. Meanwhile, the death of Mystra #2 caused the formation of Wild Magic and Dead Magic zones across Toril, which were even known as "Helmlands" because it was her death at the hands of Helm that caused them. So killing Mystra to unleash the Spellplague isn't entirely unprecedented.
Changes to Nations
Several of the most prominently magical regions of the Shining South, which WotC seemingly deemed "expendable", are burned and blasted into the magical equivalent of Fallout wastelands, collectively known as the Plaguewrought Lands. These precise areas are Halruaa, Sespech, the Golden Plains, and the Nagalands.
A portion of southern Unther is transplanted to Abeir, receiving in exchange a dragonborn city-state that becomes the new nation of Tymanther.
Large sections of Chondath and western Chessenta are replaced with a genasi-inhabited portion of Abeir, which dubs itself Akanûl.
Large portions of the Sea of Fallen Stars and Vilhon Reach collapse into the Underdark, draining away much of the water and exposing long-sunken ruins.
The shadow-tainted survivors of Netheril return from the Plane of Shadow in force and begin reclaiming their ancestral territory. Krinth and Shades of Netheril begin giving birth to Shadar-Kai, as do "shadow-touched" humans across Toril.
The long-subdued genie overlords of Calimshan (Toril's pre-Islam Arabia nation, as opposed to Al-Qadim, which is Crystal Dragon Islam Arabia) are set free. They proceed to call forth their long-slumbering legions and resumed their war, an act which also enkindled long-dormant genie blood running rampant throughout the population and caused a massive percentage to change from humans into genasi, who proceeded to conquer their human kindred.
Latan is seemingly sunk, although it's later retconned that they were just shifted to Abeir.
Dambrath experiences an uprising by a never-before-mentioned organization dedicated to restoring Dambrathian rule to the Dambrathi, overthrowing the Cintri - the Drow/Half-Elf/Dambrathi hybrid minority of Loviatar and Lolth-worshipping Amazons who've been running the place for generations.
The Feywild is brought back into planar alignment with Toril.
Mulhorand is destroyed and its people annihilated; the Deep Imaskari subsequently emerge from their long period of hiding in the Underdark and settle amongst the ruins, creating the reborn realm of High Imaskar.
Changes to Characters
In a nutshell, any human character from 3rd edition is dead and buried of old age. This includes many iconic characters like Mirt the Moneylender, King Azoun of Cormyr, and Khelben Blackstaff.
Speaking of Elminster, he loses 99% of his magical abilities and is left as a genuinely frail, embittered, tired old man.
Drizzt loses all of his companions to old age.
Tyr was manipulated by Cyric into duelling with Helm, accidentally killing the latter. In grief, he abdicates from godhood, giving his power and his faithful over to Torm instead. This titbit is actually covered in the article "Channel Divinity: Champions of Torm" in Dragon #381.
Most of the demihuman deities are downgraded to "Exarches". In practical terms, this didn't change anything, it just meant WotC declared they weren't going to bother creating feats, Paragon Paths and other mechanics to support them.