at the door：在门边
Someone is knocking at the door.
有人在敲门.
by the door：在门旁
By the door stood a man with a knife in one hand.
有个男人站在门旁,手里拿着小刀.
Who is the man that is standing by the door?
站在门边的那个男人是谁?
其实他俩区别不大
如果非要区别的话,
① 通常放在固定词组里德一定是at the door：
knock at the door 敲门
lay at the door of 归于
lay at their door 归罪于……一系列的说法都是at the door而不是by the door
② 感觉上来讲 at 有短暂接触的意味（点）；by是大面积的依着、靠着（面）
如果不确定的情况下,都用at the door肯定没错