What are Byakuya's Gokei, Senkei and Shuukei?
They are various forms of his bankai, achieved by arranging the blades in different patterns. His bankai is versatile by nature (being composed of countless tiny blade fragments), and affords him several forms, with varying combination of attack and defense. 

Whether a bankai can have multiple forms depends on its nature. Hitsugaya Toshiro and Abarai Renji could also achieve multiple bankai forms with practice/training. The former could mold Hyorinmaru's ice into various forms (like fairy-tail's Gray), while the latter could rearrange Zabimaru's segments. Shinigami such as Komamura, Tousen, or Soifon would likely not have multiple bankai forms due to the nature of their bankai.

By the way, Byakuya is no longer the only known shinigami with multiple bankai forms


   as Yamamoto Shigekuni's bankai has been revealed.


Also, Byakuya's sister, Rukia, has shikai with multiple forms:


   Tsukishiro, Hakuren, Shirafune, and Juhaku