In an ObservationContext are history information: "+History", in the ObservationContext, in level0 and in the HPSXXX or HPPXXX of the other levels. There is also an entry for quality reports:"+quality" for all observations and also "+qualitySummary" for those which have been checked by the Quality Control team at the HSC. The History contains information about the processing history of the ObservationContext or whichever layer the History belongs to. In the Quality is information added by Quality Control at the HSC or added by the SPG pipeline processing in an automatic manner. A nice way to inspect these both is to click on them (in the Data tab) and the appropriate viewer will open.
History. This contains the automatically generated script of actions performed on your data, a history of the tasks applied to the data, and the parameters belonging to those tasks. If, while in the Observation viewer, you click on the highest level +History, that of the ObservationContext itself, you will see only a few tasks listed in the table in the viewer. Go down to one of the HPSXXX/HPPXXX in the Levels (level0, 0.5 etc.) and the +History there are associated with those particular products, and so on down the various layers. This History may be useful if you want to see what tasks were run on the observation you are looking at, and what parameters were specified, however, because the History can be difficult to read (it repeats rather than overwrites itself), a more unambiguous check is to look at the "SPG" scripts in the HIPE menu for the same build as the SPG version of your observation: these are the scripts that the data were pipeline-processed with. See e.g. sec. 3.4 of the PDRG for information on these SPG scripts.. The History of any new products you create as you pipeline data are appended with your contributions.
Quality: quality and qualitySummary: Quality information is tabulated in a browser which can be accessed by clicking on either of these contexts in the Data tab.
If there is a "qualitySummary", this is what should be inspected, as it means that the observation has been checked for particular problems by a person at the HSC. As well as quality information added by the HSC, flags raised by the SPG or added automatically during the observations can be found. No comment = no problems. Where only the "quality" is present, then only the automatically-generated flags are provided.
Much information about the quality of observations and known problems can also be found on the Herschel DP Known Issues page (http://herschel.esac.esa.int/twiki/bin/view/Public/DpKnownIssues). Information about the quality report is also given in chap 6.2 of the PDD. Note that you can also get a quality report directly from the HSA: see chap 1.4.3 of the DAG.