This reaction is called a neutralization reaction. The chemical reaction for this can be written as:
NaOH (aq) + HCl (aq) ---> NaCl(aq) + H2O (l)
Generally, NaOH and HCl contain equal amount of moles thus when they reacted, there will be also same number of NaCl moles. But the concentration of NaCl produced will me lower than that of the HCl and NaOH since the total volume is increased.
The conductivity is proportional to the concentration of ions. So we cannot say that the concentration will remain the same. Let me show you an example.
1L of 1M HCL and 1L of 1M NaOH both have 1 mole of the substance.
1mole HCl + 1 mole NaOH ---> 1 mole NaCl + 1 mole H2O
1mole NaCl/total volume
1mole NaCl/2L = 0.5M NaCl
The concentration of NaCl is decreased because the total volume is increased. Therefore the conductivity will decrease as well.