Evasi0n7 is the name of the method used to apply a tethered jailbreak to the phone. The 7 is for iOS 7. The jailbreak is what disables the security features that lock people out of their own device.
TaiG is the name of a "store" the distributes Chinese applications, similar to Cydia, the store that is currently considered to be the "default" for distributing applications on jailbroken devices. Aside from using Cydia or TaiG, you can also put apps on the device manually or use other stores / distributions.
The deal with TaiG was not a result of any stealing. Evasi0n (the team that made the Evasi0n7 method) had been approached by TaiG with an offer of bundling their store instead of Cydia (which doesn't have a lot of Chinese content) for Chinese users only. Terms of the deal included that TaiG would not be allowed to distribute any "pirated" applications. Evasi0n's rational was that without TaiG on the device, most Chinese users would proceed to install an app store that did provide "pirated" apps and this way they would be condoning a "non-pirating" app store to the huge Chinese jailbreak audience. In exchange for bundling TaiG and therefore giving TaiG a huge userbase in China, Evasi0n was offered a lump of money.
Unfortunately, it turns out after the fact that some pirated apps were spotted on TaiG. Evasi0n reported these to TaiG ASAP and they were removed. You can imagine the trolling that ensued especially from competing jailbreak teams.
Other teams working on a jailbreak method in parallel to Evasi0n were also given this offer from TaiG. In fact, another team was getting a jailbreak release ready with a similar, stolen or different method, I don't know, but since they were getting close to a release, Evasi0n decided to fast-track their working method and release a jailbreak early. The up-side of an early release was that they'd get TaiG's money and they'd get the credit for the jailbreak. The down-side is that the huge volume of apps written for jailbroken devices hadn't been tested and fixed to work on iOS 7 yet, including "Cydia". iOS compatibility is even more crucial for jailbroken apps than for standard iOS apps since they often use undocumented API which is obviously very volatile across iOS versions.
As a result of Evasi0n's early release, a bunch of people jailbroke their device only to find that almost all of the apps written for jailbroken devices that they were installing crashed or cashed their phones to break - since, as I said, they weren't updated for iOS 7.
TL;DR - Evasi0n worked really hard to find a method for jailbreaking, figured they deserved some money for their effort, figured in the mean time they'd condone a safe store to the Chinese, saw their chance at success slip away as other teams were gearing up to steal the glory and released before the developer community was ready, causing breakage and mayhem, never mind the trolling about the sudden appearance of a Chinese app store instead of Cydia.
For Evasi0n's side of the story, read http://evasi0n.com/l.html