I've read the licence page of mongodb, found it under AGPL. Sorry for my poor English, I don't understand it well.
I want to build a public commercial website with mongodb, I just use it as my database, storing some data, and won't modify it. Do I need to:
No to both, this doesn't affect you, you are not redistributing mongodb or modifying the source code of it
From the license:
Note however that it is never required that applications using mongo be published. The copyleft applies only to the mongod and mongos database programs. This is why Mongo DB drivers are all licensed under an Apache license. Your application, even though it talks to the database, is a separate program and "work".
(emphasis mine)
Your website is considered a separate work, you do not forfeit any rights to it.
The blog post on the licensing extends the license page answer slightly.
Basically you have no problem. The license concerns the open source nature of the product. If you plan to modify the core database source code then you would have to report those changes back - but if you are using mongodb as a database then they do not want to know what you are doing.