As I understand it, grex refers to the parentage of a hybrid. Cultivar refers to a particular clone/strain that people have named.
In your example,
C. Gaudii (C. leopoldii f. coerulea 'Kathleen' JC/AOS x C. loddigesii f. coerulea 'Blue Sky' AM/AOS)
The hybrid is Named Cattleya (C.) Gaudii and the grex (the parentage) is given in parenthesis. The cultivar names of the parents are given in quotes. This particular hybrid, C. Gaudii, was made by crossing two select parents, Cattleya leopoldii and Cattleya loddigesii which were both apprently named and awarded cultivars. The JC/AOS and AM/AOS refers to the awards the plants have received.
Some species have variation of color and or pattern on the flowers that are a bit different than what's considered "typical". These are sometimes indicated by f. (forma) or var. (variety). There are scientific distinctions on those but I'm not a taxonomist, so I will leave that alone. In your example, the parents are both f. coerulea, or a lavendar/blue-violet like color variation of the species.
The AOS website has a nice introduction here:
Basics of Orchid Names
For me, it is nice to google for a photo of the hybrid - or if the hybrid is not very common, looking at the parents and imagining what the offspring (hybrid) might look like.
google, facebook, and orchidroots.com are a great resource for browsing!