First of all, you can no longer pick whoever you want as a rival. Before people tended to use the rival system as a targeting mechanic, if you wanted to fight a weaker party you set them as rival. Now we have a picking mechanic, which incidentally is the same as when the AI picks rivals, that gives you a selection of nations that are serious competitors. In this list there should not be nations that are significantly stronger or weaker than yourself, also things like being a naval power plays a part.
Then, to measure how well you are fighting your rivals (if you name someone your official rival, if you fail to fight him you will look weak) we add Power Projection value. Power Projection is a value that gets added to for things you do to fight your rivals. For example you get Power Projection if you are embargoing a rival, if you send a warning to your rival or supporting rebels in a rivals nation. Another interesting example is that you get power projection for sending money to nations your rivals are fighting in a war. Those things I just mentioned are static and gives you positive power projection while they hold true. Then you have some dynamic sources of Power Projection that gives you a one time boost that will then decay over time, like a boost when you take provinces from a rival or when the rival loose a province to someone else in a war where you participate. You can lose power projection as well, you loose power projection for example if you lose provinces to your rival. The Rivals system is supposed to reflect actual rivalries in the world and bigger nations are expected to have more rivals (max 3 though) and if they have less than that their power projection will suffer. You can also not just change rivals whenever you feel like it, you can't change rival for 10 years after you picked it and it carries a significant Diplomatic Power cost, so think carefully about your choice. You can however not go under zero, so you would like to continue to ignore your rivals in the world, feel free to do so. If you on the other hand show the world that you can stand up to your rivals, you get increased defensiveness, trade power, military morale and legitimacy. And if you really manage to be active against your rivals you get the most interesting reward, if you have over 50 Power Projection you will get +1 monarch point of each type every month.