If something's pressing, it's very important and should be taken care of right away. Your French club's most pressing issue might be raising money for the big trip to Montreal.
Pressing business needs urgent attention, and pressing problems will turn into disasters if you don't figure out a quick way to solve them. Another meaning of pressing is "the act of exerting a strong pressure," like the pressing of apples into delicious cider. Pressing comes from the verb press, "to push against" or "to squeeze out," and later also "to urge or argue for."