What happens to non-believers or believers of other religions after death? What happens (according to Christians) to people who specificially rejected the idea of God, the Bible and the christian teachings after they die? 

Is there a difference to people who believed in another deity, maybe even several deities?

We have seen questions what happens to people who have never heard about Jesus, unborn babies and children. I did not see that question before, so I asked right away. Additionally I don't ask if we can go to heaven, but simply what happens after our life has ended.
The Bible is very clear that all atheists and non-believers will be sent to the Lake of Fire after they die.  It even goes further than that by specifying a list of people that are not accepted (which covers more than your question!):


  Revelations 21:8 (NIV)
  But the cowardly, the unbelieving, the vile, the murderers, the sexually immoral, those who practice magic arts, the idolaters and all liars—they will be consigned to the fiery lake of burning sulfur. This is the second death.


To answer the question, it's pretty clear that the Bible does not distinguish between "non-believers" (atheist, agnostic, or what have you) and "believers of other religions".  When it says "the unbelieving" it refers to all non-Christians.

Furthermore, it's the belief in God that brings salvation, not the works.  We can see from this verse that it's what's in a man's heart that determines his salvation.  Every one of these evils can be directly related to a problem in the person's heart.


  Acts 16:31 (NIV)
  They replied, “Believe in the Lord Jesus, and you will be saved—you and your household.”


However, works are an outpouring from faith:


  John 14:15 (NIV)
  If you love me, keep my commands.


And without the faith that produces works, there is no actual faith behind it:


  James 2:14-18 (NIV)
  14 What good is it, my brothers and sisters, if someone claims to have faith but has no deeds? Can such faith save them? 15 Suppose a brother or a sister is without clothes and daily food. 16 If one of you says to them, “Go in peace; keep warm and well fed,” but does nothing about their physical needs, what good is it? 17 In the same way, faith by itself, if it is not accompanied by action, is dead.  
  
  18 But someone will say, “You have faith; I have deeds.”   
  
  Show me your faith without deeds, and I will show you my faith by my deeds.  


Ultimately, to answer the question, all non-believers (people who do not believe in any god, the Christian God, or who believe in another God) along with all people who claim to believe in God but do not have a solid faith will end up in the lake of fire.