Signalling is the idea (originally from biology) that your actions convey information about you to others. A red mushroom is signalling to would-be predators that it is poisonous. However, not all signalling is honest signalling. Once an association between the colour red and bad consequences exists, other species are incentivised to dishonestly signal poisonousness (and not get eaten or have to spend resources making the poison).
Countersignaling, on the other hand, is showing off by not showing off. For example, imagine you are so rich that signalling your wealth is no longer worth the effort, so you don't splurge on fancy cars and extravagances. Your lack of signalling wealth is countersignalling that you are wealthy.
Examples of signalling and countersignalling
Examples abound in the animal kingdom (like springboks’ stotting or peacock’s tails), but what about in humans? Yep, that too. In work that won Michael Spence a Nobel Prize, he made the case that potential employees signal to get ahead in the job market. They go through costly (and therefore credible) education to prove that they will be valuable employees — because if you can get through years of learning unnecessary content, work will be a cinch (or so the cynical argument goes).
Buying an expensive wedding ring signals your intentions not to do a post-coital dash (lest you lose three to six months of your salary).
This theory predicts that the nouveau riche spend more to signal their wealth (fast cars, extravagant houses, fine dining), compared to old money, who can get away with more countersignalling.
Also check out
What is signalling really?, Scott Alexander
Friendship is Countersignalling, Scott Alexander
Job-market signalling, Wikipedia