the first time 第一次,是名詞,在句子中只能做名詞的成分.
at the first time 是介詞短語,是“在第一次”
for the first time 也是第一次,是個副詞短語,在句子中做狀語.
例子：
I felt nervous the first time I gave a speech to many students.
我第一次給學生做演講時感到很緊張.
He won the match at the first time.
他在第一次比賽中就獲勝了.
I felt repentance and sorrow for the first time .
我第一次感到后悔,感到難過.