In classical physics, we have second-order equations like Newton's laws, so we need to specify both position (zeroth order) and velocity (first order) of a particle as initial conditions, in order to pick out a unique solution.
In non-relativistic quantum mechanics, we have Schrödinger's equation, which is first-order. As initial data we can therefore choose only the wavefunction's value at each point in space, but not its time derivative. This seems to dovetail with the uncertainty principle, which says a quantum particle does not have independent position and momentum degrees of freedom. We can choose the wavefunction to specify either a position or a momentum, but not both. (Or we can also choose a wavefunction that has uncertain position and momentum, within the bounds of the uncertainty principle.)
In quantum field theory of fermions, we have the Dirac equation which is again first-order like Schrödinger's. But not every quantum particle is a fermion, and, AFAIK relativistic non-fermion particles obey the Klein-Gordon equation, which is second-order!
So with the Klein-Gordon equation we can apparently choose both the wavefunction and its time derivative at each point in space, giving more freedom than the Schrödinger equation. Why do we have this extra freedom, and how can it be reconciled with the uncertainty principle as applied to relativistic bosons?