What exactly is the time step in weather predictions? The question basically says it all. I am confused as to what the 'time step' in weather prediction signifies-- is it the time interval between each measurement of the initial condition? Or is it the time interval used to define the prediction itself? If it is the latter, will the resolution of the grid model affect the time step?
 A: Weather predictions are based on strongly nonlinear differential equations which are integrated numerically. There is a great chance that the question is referring to the time between two successive steps of the integration algorithm that "predicts" the system evolution.
It is said that Lorenz discovered chaos while messing with some decimal figure of a parameter in his weather prediction equations. The strong nonlinearity of the system makes it extremely sensitive to the given initial conditions; therefore, the integration time step is very important in the accuracy of the prediction of the system evolution.
A: The time step is the latter - it is the time duration between which all the variables of the atmosphere are calculated by the model. These calculations are performed by solving a linked set of partial differential equations, known as the governing equations of the model. 
For example, if you know the temperature, pressure, wind vector at all point on a 3D grid at time $t$, you can predict the values of these variables $t + \Delta t$. Here, $\Delta t$ is the time step. This is done by solving a discretized (in space to the 3D grid, and in time to the discrete times) version of the governing equations. 
There is a link between the spatial resolution (length of grid-cell - $O(1\ \mathrm{km})$ for NWP) and temporal resolution (time step - $O(60\ \mathrm{s})$). It depends on exactly how you choose to discretize the governing equations in time, but normally the spatial and temporal resolutions are linked through the CFL condition. Essentially, this states that as you go to higher spatial resolution, you must also go to higher temporal resolution, or your simulations will become unstable.
Note, the above is a simplification. Most NWP models would have different time steps for different parts of the model. For example, the dynamics (how the winds advect round other variables) might be solved using a short time step, and the radiation (which is computationally expensive) might be solved using a longer time step that is a multiple of the short time step.
How you actually get observations into an NWP model is a whole different kettle of fish! The times of the observations will not correspond to the time steps of your model. You must account for this, and try to come up with some best guess of what the actual state of the atmosphere is, given previous predictions from you model and the observations. In atmospheric science, this is known as Data Assimilation.
