> I am not a physicist and I am certainly in dangerous half-knowledge territory here.
Gauging is just dealing with the fact that there is no absolute universal fixed value against which can compare a value at some point in a field; but we still want to consider values at one or more points in the field.
Let's do a really simple static model of the atmosphere, with a single scalar value: air pressure at each point. Let's use a simple device: an air pressure gauge which reports some fraction of a pressure measured when we push a "calibrate now" button. We'll call this a calibrated barometer. We can then recover the full air pressure field by measuring at every point in space (not space-time, the staticity means there is no time-dependence to the measurements; we can do them in any order and not have to worry about time of day or season).
Where do we push the "calibrate now" button? At some point on the surface? At mean sea level? At the top of the atmosphere? The choice of any of these will provide different readings on our gauge (i.e., it reports some fraction of the calibrated pressure, will differ when the calibration point is 101 kPa vs some fraction of the value actually measured at a specific point on the surface). But with a bit of care in choices of units, whatever we use as the calibration point, the difference between two different points in space will be the same.
A good choice of gauge lets use our calibrated barometer as an altimeter. In aviation, aircraft pressure altimeters have a calibration knob, which is used to recalibrate during different stages of a flight. Common calibration points are: QFE, field elevation, which lets one know how far above an airfield one is if separated only vertically from it, at the cost of being unable to simply compare the vertical separation between two aircraft above two different airfields; SPS (the pressure of the standard atmospheric pressure, 1013.25 hPa) is a global setting useful for quickly determining the vertical separation between two reasonably nearby aircraft, at the cost of not being able to quickly determine height above terrain, or even the height above mean sea level; QNH is another local setting which lets one compare how high above mean sea level the aircraft is, at the cost of needing to know the height above mean sea level of local terrain, and not being able to easily compare vertical distances with aircraft using one of the other two calibrations.
All three settings are "redundant descriptions" of the aviator's atmosphere. They describe the same column of air, but each makes it easier to pinpoint different hazards scattered through that column (the ground, other aircraft in level flight, the aircraft's operating ceiling).
We could complicate the atmosphere by introducing time dependency (at night in cold dry winter a QNH altitude will be fewer RADAR-measured metres above the same patch of ground), and atmospheric interactions (atmospheric waves, Bernouilli effects from winds). Each complication can be made to vanish via a careful choice of gauge, although it gets harder and harder to write down such a gauge as complications increase. (As a result, in aviation they allow for a certain amount of measurement error and safety margin, and comparisons with different means of measuring altitude like radio altimeters and satellite multilateration.)
In a quantum field theory (QFT), one might choose a gauge in which some particles vanish. A sibling comment pointed out that very commonly one wants to choose a gauge in which gauge bosons like photons don't need to be counted, rather than a gauge in which there is a sea of an enormous number of low-energy gauge bosons. Choosing the gauge does not eliminate the low-energy gauge bosons; in general QFT field values are time-dependent (and usually gauged to admit only "relevant" fluctuations). Low-energy fluctuations can be boosted into "real particles" by relativistic observers, and strongly accelerated observers can count more particles than a weakly accelerated one. Therefore the choice of a gauge for one observer might make calculations for another observer more difficult.
In QED there are several well-known and frequently-used gauges roughly analogous to SPS/QFE/QNH, and one often chooses one of them for convenience. Each of tehse gauges breaks the gauge freedom.
Gauge freedom means simply an uncalibrated system waiting to be calibrated. A common illustration of this is to choose a non-rotating sphere and setting down latitude/longitude. A less-gauge-symmetrical rotating sphere naturally picks out latitudes (the poles and the equator, notably), but there's still gauge freedom in longitude that we can fix by choosing a prime meridian, and gauge freedom in picking out one of the primary compass directions. These choices do not change the sphere or its rotation (or non-rotation), and of course one can choose any other set of coordinates one wants.
Once one has fixed the gauge on the sphere, though, one can more easily compare positions on the surface: is point A in the northern hemisphere, is point B in the eastern hemisphere? Just asking if point B is North-East of point A requires us to at least choose a north pole -- that can be one of two places on a rotating sphere, and it can be anywhere at all on a non-rotating one. The "right hand rule" is the conventional "gauge" for rotating astronomical bodies: anticlockwise rotation around the north pole (right hand: thumb up, fingers curled). But we don't have to use that convention as our "gauge". (We also have a problem for a truly non-rotating spherical object: where's the north pole? We might solve that by using an imagnariy axis parallel to the axis of a relevant body like the local star or the parent galaxy).
Finally, in many gauge theories there are gauge invariant quantities. On our spheres the geodesic intervals between two points are gauge invariant. The gauge tells us something about direction. In practice, fixing a gauge also usually involves choosing (and scaling) units: on our geodesic which might run south-east to north-west (gauge problem), the length might be measured in metres or light seconds (units problem) or kilometres and light-years (scaling problem). We might want to label different points along the geodesic in latitude/longitude (coordinate problem) rather than adapted Cartesian sphere-centred/sphere-fixed ("ECEF" on Earth) or tangential ("Local East-North[-Up]", "LENU") ones.