When used with respect to survey control, order is a specification of the relative accuracy of the positions of stations in a geodetic control network. The NGS has standards that must be met for the different orders of accuracy.
On an NGS datasheet, one of the lines will say "HORZ ORDER" (geocaching datasheets do not have this specification line). The levels of order and their accuracy are:
GPS Standards 

Horz. Order 
Definition 
AA 
a relative accuracy of 3 mm +/ 1:100,000,000 relative to other AAorder stations. 
A 
a relative accuracy of 5 mm +/ 1:10,000,000 relative to other Aorder stations. 
B 
a relative accuracy of 8 mm +/ 1: 1,000,000 relative to other A and Border stations. 
Classical standards 

Horz. Order 
Definition 
First 
1: 100,000 relative to other stations of the same order. 
Secondorder, class I 
1: 50,000 relative to other stations of the same order. 
Secondorder, class II 
1: 20,000 relative to other stations of the same order. 
Thirdorder, class I 
1:10,000 relative to other stations of the same order. 
Thirdorder, class II 
1:5,000 relative to other stations of the same order. 
The classical relative accuracy specifies a distance tolerance. For instance, if two firstorder points are separated by 10km, then the distance between them will be accurate to 10 cm, in principle. To estimate the absolute accuracy of a position, a reasonable estimate for a "classical" station would be based on the distance from the station to the nearest CORS, HARN, or FBN station that existed at the time of adjustment within the same state as the station. For instance, here is a portion of a datasheet:
LY2625 DESIGNATION  BEVANS LY2625 PID  LY2625 LY2625 STATE/COUNTY NJ/SUSSEX LY2625 USGS QUAD  CULVERS GAP (1997) LY2625 LY2625 *CURRENT SURVEY CONTROL LY2625 ___________________________________________________________________ LY2625* NAD 83(1996) 41 12 07.74590(N) 074 51 22.66670(W) ADJUSTED LY2625* NAVD 88  235. (meters) 771. (feet) SCALED LY2625 ___________________________________________________________________ LY2625 LAPLACE CORR 0.90 (seconds) DEFLEC99 LY2625 GEOID HEIGHT 32.54 (meters) GEOID03 LY2625 LY2625 HORZ ORDER  FIRST LY2625 LY2625.The horizontal coordinates were established by classical geodetic methods LY2625.and adjusted by the National Geodetic Survey in September 1999.
The nearest highaccuracy station in New Jersey in 1999 was LY2920, located 25 km away. That would make the absolute position of LY2625 accurate to about 0.25 meter.
A forthcoming adjustment of all GPS stations, scheduled for completion in 2007, will list absolute accuracies on the datasheet. Unfortunately, classical stations will be excluded from that adjustment, so estimates such as the above will still be necessary.