From the wikipedia entry:
Double bonds: E/Z
For alkenes and similar double bonded molecules, the same prioritizing process is followed for the substituents. In this case, it is the placing of the two highest priority substituents with respect to the double bond which matters. If both high priority substituents are on the same side of the double bond, ie. in the cis configuration, then the stereoisomer is assigned a Z or Zusammen configuration. If, by contrast they are in a trans configuration, then the stereoisomer is assigned an E or Entgegen configuration. In this case the identifying letters are derived from German for 'together' and 'in opposition to', respectively.
Although, thinking about it, perhaps I understand the source of your confusion.
If one were to have the di-bromo structure below, that would be
E or
trans.
But if one were to have the dibromo-dichloro structure, then this would not be assigned as E/Z, but rather 1,1-dibromo, 2,2-dichloro. I think I'm right with this, but perhaps someone could confirm...?