What is the difference between a high-resolution model and a regular model run?
High resolution means that the grid mesh the model uses is fine, or put another way, $dx$, $dy$, $dz$, and $dt$ are small numbers.  A model like the HRRR uses a horizontal resolution of 3 km, whereas some of the older regional models used horizontal resolutions of 30 km (though even the NAM is using 5 km these days).  Comparing resolutions to GFS and ECMWF models is a bit less straightforward because those models solve in a spherical coordinate system using spectral methods, and resolution will vary with latitude).

Higher horizontal grid resolution also implies higher vertical resolution and a smaller timestep.  The vertical resolution will be a consequence of wanting to avoid large aspect ratios in your grid box and the smaller timestep follows from the need for numerical stability (see CFL criterion).  The biggest direct result of the increased resolution and decreased timestep are increased memory and drastically increased computational requirements.  Because of this, operational forecast model resolutions will be limited by how long is acceptable for the model to run before output is available, which is a function of the supercomputer crunching the numbers.  You will generally find much higher resolutions in non-forecast research settings than operationally.  For example, I simulate supercell thunderstorms with 250 m horizontal resolution, but I can also spare an entire day to simulate a few hours whereas an operational model only has a few hours to simulate many days in a much larger domain than I use.

I don't believe there is a specific requirement to call a model "high resolution" and I think you'll find that the implied resolutions will be quite different between a climate modeler and a mesoscale modeler.  

Differences in accuracy in higher resolution models generally boils down to resolving more of the physics explicitly.  A model running with 30 km resolution cannot resolve convection (thunderstorms), and must parameterize this as a sub-gridscale effect.  You might see terms like "cloud resolving" applied to high resolution models because of this.  A model running at 3 km resolution will be able to explicitly resolve convection.  A large eddy simulation (using filtered Navier-Stokes equations) will be limited to resolving eddies on a scale based on the grid resolution, and will parameterize the smaller scales.  Please see this paper1 for further discussion on the impact of resolution on resolving deep moist convection.

Finally, the data going into the model will make a difference (weather is chaotic and extremely sensitive to the initial conditions).  The HRRR is initialized from the RAP with a data assimilation period for observed radar.  RAP is currently a 13 km model but is moving to 3 km this summer. RAP gets its background fields from GFS and assimilates quite a bit of data on top of that.  While the smaller grid boxes help capture more of the physics, the quality of the data fed into the model is also helping quite a bit with accuracy because it improves the initial conditions.  




Bryan, George H., John C. Wyngaard, J. Michael Fritsch, 2003: Resolution Requirements for the Simulation of Deep Moist Convection. Mon. Wea. Rev., 131, 2394–2416.
doi: http://dx.doi.org/10.1175/1520-0493(2003)131&lt;2394:RRFTSO>2.0.CO;2