From what I could figure, you are in the case of circular motion with constant acceleration see Wikipedia
Be careful with your notation, because in this case there are 2 velocities, angular and linear. Angular is independent of the radius (usually denoted as $\omega$) while linear is actually $\omega\cdot r$.
So, probably in your first equation $\theta=\omega\cdot t - \frac{1}{2}a t^2$ you are mixing linear and angular quantities.
In any case, to answer the last part, $a$ is the acceleration, i.e. second derivative with respect to time of the angle $\theta$, $\frac{d^2\theta}{dt^2}$, and describes how the angular velocity changes with time. The $\frac{1}{2}at^2$ is a straightforward derivation from the equations of motion for constant angular acceleration.