'Taeyeon palagi' literally means 'Taeyeon flower', it was created when the word 'Sunflower' in Korean, 해바라기 (hae palagi) is altered.
When 'Taeyeon' replaced the 'sun', it just means that the person always looks at Taeyeon, thinks of her and follows her around, equivalent to Taeyeon bias. Short form for it (given by the girls) was 태바 (Tae-ba, sounds just like 'Dae-bak' which is really daebak.)
Similar rules apply for all other members:
태연바라기 (태바) Taeyeon-palagi (Tae-ba)시카바라기 Sica-palagi (시바/싴바 doesn't sound good, please refrain from using it)써니바라기 (썬바) Sunny-palagi, Sun-ba (Also can be 순규바라기, 규바 Soonkyu-palagi, Kyu-ba)미영바라기 (묭바) Miyoung-palagi, Myong-ba효연바라기 (효바) Hyoyeon-palagi, Hyo-ba유리바라기 (율바) Yuri-palagi, Yul-ba수영바라기 (셩바) Sooyoung-palagi, Soo-ba윤아바리기 (융바) Yoona-palagi, Yoong-ba서현바라기 (서바/현바?) Seohyun-palagi, Seo-ba or Hyun-ba
credit: @imwhywhy
Ci-Luk-BAAAA~~~