In the reaction of nitrogen and hydrogen how is ammonia produced even though the reaction is reversible?

We get ammonia because the forward and reverse reactions are happening at the same rates.

If 3 mol of #H_2# is mixed in a sealed vessel with 1 mol #N_2# under suitable conditions then they will react to form ammonia #NH_3#:

#N_2+3H_2rarr 2NH_3#

At the start of the reaction the concentration of the #N_2# and #H_2# are high. As soon as some #NH_3# is formed the reverse reaction will start to occur:

#2NH_3rarrN_2+3H_2#

The rate of the reaction depends on concentration so the forward reaction will be fast at first when the concentration of the reactants is high. It will slow down as their concentration decreases.

By the same reasoning the reverse reaction will be slow at first then increase. These two processes continue until a point is reached when the rates of the forward and reverse reactions are equal.

We now state that the reaction has reached equilibrium which we show by:

#N_2+3H_(2)rightleftharpoons2NH_3#

It is described as "dynamic" because the forward and reverse reactions are happening at the same time although the concentrations of all the species remain constant.

So although #NH_3# is constantly breaking down, more is being formed at a constant rate.

In the Haber Process the system is actually not allowed to completely reach equilibrium as the process is continuous, as described in Mukhtar's answer.