1. open the door 与 answer the door：前者表示一般意义的“开门”，与“关门”(close the door)相对；后者指听到门铃响或敲门声而去开门。如：

He opened the door and went out. 他打开门就出去了。

Will you answer the door? There’s someone knocking. 请你去开门，有人在敲门。

2. show sb the door 与 show sb to the door：前者意为指着门要某人出去，即把某人撵走或下逐客令；后者意为送某人到门口，即相当于汉语中的送客。如：

After a long argument he showed me the door. 争论了好一阵之后，他向我下了逐客令。

Mr Brown is leaving now. Will you show him to the door? 布朗先生要走了，你送他去门口好吗?

【注】show sb to the door 也可说成 see sb to the door。如(from www.yywords.com)：

Goodbye, Mr Carter. My secretary will see you to the door. 再见，卡特先生，我的秘书将送你到门口。