the first time 第一次,是名词,在句子中只能做名词的成分.
at the first time 是介词短语,是“在第一次”
for the first time 也是第一次,是个副词短语,在句子中做状语.
例子：
I felt nervous the first time I gave a speech to many students.
我第一次给学生做演讲时感到很紧张.
He won the match at the first time.
他在第一次比赛中就获胜了.
I felt repentance and sorrow for the first time .
我第一次感到后悔,感到难过.