I think it's per seconds, or more likely per minutes for an individual peak. We plot (relative) abundance against retention time. That said, you then use the peak area (or peak height) of your unknown WITH that of a standard. In other words, you use a ratio which has no dimensions.
You mention calibration, This is simply running a known concentration of a known compound through your system. Your software will usually be able to fit your unknowns to this in terms of retention times. You may get very reproducible retention times on a sample but unless you calibrate properly, how do you know that your retention time is accurate? Calibration compounds can be traced right back through suppliers & manufacturers to a gold standard method..which is how they, the callibrator, is proven.
It is worthwhile also running (internal) quality control samples. These tell you if your settings are still correct (old columns can degrade, crack etc). These will test the accuracy and precision of your system.
Clinical systems usually subscribe to external quality systems as well which show how they perform against other methods.
Hope that helps.