"shuffled off this mortal coil" is Shakespeare's poetic way of saying "died".  To "give pause" means to make you stop and think about something. At this time in the play, Hamlet is considering suicide and in this speech he is speaking his thoughts aloud. The part that contains the line line you quoted is : 

To die, to sleep--

To sleep--perchance to dream: ay, there's the rub,

For in that sleep of death what dreams may come

When we have shuffled off this mortal coil,

Must give us pause

'Perchance' means perhaps or possibly.  'There's the rub' means 'that's the problem'.

So, to paraphrase this part of the speech in more modern English:

To die, to sleep.

To sleep, perhaps to dream. That's the problem.

For in that sleep of death the things we dream of

After we have died

Should make us stop and think.

Just as an aside, this kind of speech where an actor in a play is speaking his thoughts aloud to himself rather than talking to another character is called a soliloquy. This speech (starting from 'To be or not to be') is usually referred to as Hamlet's soliloquy.

Thx Gareth, I really appreciate your explanation.