Nominative, subject case:
  1st person singular:    I rejoice.
  2nd person singular:    You moan and groan  
  3rd person singular:     He is lazy  
  She is energetic  
  It is important.
 
  1st person plural:   We work  
  2nd person plural:     You rest.
  3rd person plural:     They sing and dance.
 
In English we have to use pronouns to know who the subject of the sentence or phrase is, i.e. who is or does what. In Greek we don't, unless we want to emphasise the fact that it is:
I personally, I for one, I most certainly do this.... Iin which case we use ἐγώ
You (and no one else, not me, not him....) in which case we use σύ
So I rejoice while you moan: ἐγὼ μὲν χαίρω, σὺ δὲ στενάζει.
And in the plural: We work, you rest: ἡμεῖς μὲν πονοῦμεν, ὑμεῖς δὲ ἡσυχάζετε.
 
Greek Usage:

In the nominative case we only use pronouns for emphasis. But what happens in the other cases? How do we say: me, yours, us, your, them etc.?

1st and 2nd person plural are used the same in Greek as in English. But in the singular ancient Greek has 2 forms:
  • one emphatic in speech and therefore accented in writing (to reflect this oral emphasis)
  • one neutral, unemphatic in speech and therefore unaccented and enclitic in writing.
     
    Click here for those forms.
As for 3rd person pronouns, go here to find out how the Greeks dealt with those.