The term "online" typically refers to event reconstruction, and "offline" to analysis of the reconstructed events. The reason they are called "online" and "offline" is that online reconstruction is performed in real time as data is being collected, and "offline" is performed later using resources (computing banks, storage, etc.) that are not directly tied to data collection.
I'm not sure I'd use the term "online analysis", because it isn't really analysis. "Online processing" is more accurate.
To understand online processing better, it's important to realize that a high energy physics detector is not actually one detector; it is in fact a loose federation of many subdetectors that function somewhat independently. Each subdetector has its own unique processing and storage capability that collects and temporarily holds data from each event. There are online systems (triggers, filters) that examine data from each of the subdetectors and decide whether the event is real and sufficiently interesting to keep. Because there is so much data and so little time, this is done in several stages. The first stage is typically implemented in hardware (for speed); it looks at output from a few of the subdetectors - including, in some cases, detectors that are special-built for just this purpose - and decides whether the output from all the subdetectors should be read out or whether it should be discarded. If the online processing makes it past the first stage, it moves on to the second, where data from more of the subdetectors is processed and a similar keep-or-toss decision is made. As it moves through the stages, more processing is performed, data from different subdetectors is merged and compared, and similar decisions are made. Once it passes the final filter, the final stage of event reconstruction is performed where all the (pre-processed) data from all the subdetectors undergoes final online processing and is merged into a single "event", which is then stored for offline processing.
Similar processing can be (and often is) performed offline using stored "raw" data; this is usually done to incorporate improvements in calibration and to reflect a better understanding of the subdetectors' operation. Again, I'd call this "offline processing" rather than analysis.
Offline analysis is the process where physicists examine the event data -- or, more accurately, sift through millions of events and select the ones that meet certain criteria -- and try to figure out what caused the event. This is where the interesting physics results come from, such as discovery of the Higgs, or precise measurements of certain branching ratios, or limits on supersymmetic parameters, or any of the other myriad results published in physics journals.