There's alot of miss-information by tmad40blue, but the gist of it is correct. To clearify what actually happened:
The original maintainers of bukkit, after spending 3+yrs on the project, decided to end the project for a number of valid reasons(EULA, personal strife, etc), so they made a post stating that bukkit would not be updated to 1.8. One of Mojang's Employees then stated in a tweet that "bukkit belongs to Mojang", and "I'll personally be updating bukkit to 1.8". Which is what pissed off the very involved developer Wolvereness (amoung many other contributors to bukkit/craftbukkit). He made a DMCA against bukkit/craftbukkit, spigot and others that based their code on the craftbukkit project.
This all played out over the course of a month, during which the Minecraft community roasted Notch(the largest shareholder of Mojang) for something he hasn't been involved in for 2yrs. He got tired of the BS and sold his share to Microsoft.
Spigot has risen as the craftbukkit replacement due to being the only drop-in replacement for bukkit when the project faltered. With that said, its far from being as comparable to vanilla as craftbukkit was. Before bukkit fell, spigot aimed at server-side performance; to do such they took leisure with mechanics such as mob AI, and redstone ticks. Furthermore, the spigot project's lead maintainer is not, errr, open to fixing failings of the Bukkit API nor is he willing to give up the mentality of "High performance MC server!" So if you expect for vanilla gameplay it may be better to look elsewhere which is why core members of the spigot team have been working on a new project called Paper Spigot
Sources:
EvilSeph discontinues bukkit project
Playout between EvilSeph and Mojang
Archive of Wolvereness DMCA - Offending pages have been deleted so had to resort to an archive
Yep. When Microsoft acquired Mojang in August 2014, it was revealed that they had owned Bukkit for over 2 years. Obviously all the people who had been contributing code to the Bukkit project weren't so happy about their contributions being put into the game and sold without their knowledge, so they put forth a DMCA on all the Bukkit code and attempted to take down the project. In response, Mojang privatized and subsequently deleted all of the Bukkit repositories, and made it illegal to redistribute the Minecraft server software in any way (Bukkit included).
The Spigot team then swooped in and took over. Nowadays you're required to build Spigot on your own local machine through BuildTools, which basically downloads the vanilla Minecraft server JAR from Mojang, decompiles/patches/does magic to it, and builds Spigot right there locally. They also maintain Bukkit/CraftBukkit as a convenience, but Spigot is definitely the king (since it has tons of bugfixes, improvements, and performance tweaks) and is 99.99% backwards-compatible with all Bukkit plugins.
The Spigot team has been responsible for the last several official Minecraft updates because they find and fix security flaws months faster than Mojang does.