IIR initial condition / state meaning
The states depend on the form of an IIR filter. Take second order filters for simplicity. The following picture shows the structure of a transposed direct form II filter.

The differential equation is
$y(n) = b_0  x(n) + s_1(n)$
and the filter states can be calculated by recurrence
$s_1(n) = b_1 x(n-1) + s_2(n-1) - a_1 y(n-1)$
$s_2(n) = b_2 x(n-1) - a_2 y(n-1)$
As for (transposed) direct form I, it should have a filter states with 4 elements. The following is the transposed direct form I implementation of a second order filter.