First off, glad to have you here! Awesome to see new faces streaming in.
There are a couple answers to your question.
The functional/"What does this mean for my weather?" answer (what I'm assuming you're looking for) is that troughing is associated with a southward "dip" in the jet stream+storm track; the inverse of a "ridge", which is a northward surge in the jet. Troughing is (usually!) associated with cooler, wetter weather and is what us snow weenies look out for in the winter, since they can bring down arctic air from Canada. There is a more technical answer to this question, since troughing is a real meteorological term that is associated with physics; although you do not need to learn this, since local weather knowledge+pattern recognition goes a long way, and is in most cases all you really need.
If you are interested, the more technical/"correct" answer to this involves the literal thickness of the atmosphere. Warm air being less dense than cold air means that it literally takes up more volume per unit mass comparatively. This is significant because we know that due to gravity, the atmosphere exerts a certain amount of pressure at any given point on or above earth's surface. We can measure atmospheric pressure using a device called a barometer, which uses mercury to determine just how heavy the atmosphere is at that location; usually measured in millibars (mb) due to how fine that unit of measurement is. The higher you go, the less pressure is exerted, since there is less atmosphere weighing down from above (the rest is below.) At sea level, where most of the world lives, the atmosphere exerts roughly 1000mb of pressure.
For the sake of simplicity, let's say we have two identical airmasses, except one is 5°C cooler all the way up (surface to tropopause) than the other. We'll call the cooler one our "trough" sample. Now since we know that cold air contains less volume than warm air due to its higher density, we can infer that our trough airmass will be denser, and consequentially less spacious, than our warm airmass. To understand why that's important, let's compare what's going on inside our two examples. Starting at the surface, in this case sea level, we find the same measurement between the two (close to 1000mb), since in both cases the same amount of mass is weighing down from above.
But as we increase our elevation, at the same rate, we'll find that the amount of atmospheric pressure exerted begins to diverge. We'll notice that the colder airmass will have a lower atmospheric pressure than the warmer airmass, even at the same altitude, let's say 15,000 feet. In other words, we do not need to go as high up to find a given pressure reading than in the warm airmass. The elevation at which, say, 500mb exists, is literally lower in a cooler airmass than in a warm one. If you were to make a cross-section, with x representing a slice of the surface and y as altitude, you'd find that there would be a "dip" in 500mb elevation where the airmass was cooler. Hence, a "trough"!
Why is this? Well again, we need to think about our airmasses in terms of both temperature and volume. As we raise our barometer higher into the trough airmass, we'll find that we're rising above more air molecules at a faster rate than in the warm airmass, since the air is cooler and denser. To get a better idea of why this is, I threw together a diagram, since I am not too elegant with words. The same concepts are there, but presented visually. Essentially the term trough in an atmospheric context is used when thinking in terms of pressure, like a topographical map. Wikipedia has a great article on it: https://en.wikipedia.org/wiki/Trough_(meteorology).