The electrode compartment

is the space directly around an anode or cathode, its surface included.
There the half reactions of the indirect redox reaction take place.

The Cathode and anode compartments may not be separated physically, and everything takes place in one solution or liquid.
To keep the situation equal in all places, stirring of the solution is needed continuously.

Another option is: the two processes occur in separated compartments, but connected with semiporous barriers of with salt bridges.
Without one of these, there is no closed circuit and current is impossible (nor any transfer of electrons)
In the case of a salt bridge, take care that the ions in that salt bridge themselves do not participate in the redox reaction.

A choice pro or contra separated electrode compartments has all to to with what you aims with the redox reaction in charge.
Keeping apart mostly aims to keep de products separated.
These product could possibly react (when in contact), and - for example - produce a precipitate or a gas.
Onthe other hand, someone can also wish that happen.

A solution (or a molten substance) conducts electricity only under the condition of the presence of charged and movable particles (ions in this case).
The conductibility has everything to do with the movability of those ions: The more movalbe, the better they conduct.
    That movability depends on some factors:
  1. how many ions are present?
  2. are these ions big or small or hidrated?
  3. Somtimes an ion (certainly the H+-ion in water) has special techniques to improve their movability.