Radars and sensors are implemented pretty weirdly in Arma, with the two of course being independent systems but the radar is overlayed on top of the sensors essentially making the usage of both identical for better or for worse. Below is a summary sheet and some light notes further down.
Ideally you'd go through the built-in showcases to figure out basic controls for the radar/sensors, but reminder that R can cycle through targets on the radar/sensor and T will target whatever you're currently looking at (if anything is there).
The active radar is the radar you can turn on and off, and the one that you can see with the blue overlay. You need the radar to be enabled to use MR/LR Air-to-Air missiles but bear in mind, having your active radar on does reveal your position to RWR and/or passive radars.
RWR stands for Radar Warning Receiver, usually it would be used to warn you if you're being locked by an an enemy regardless of the direction they're coming from. A passive radar receives oncoming pulses and can detect vehicles that have their active radar on, while not exposing itself via radar pulses itself.
The infrared sensor would be used for SR heatseeking air-to-air missiles and/or heatseeking air-to-ground weapons if you're using them. It doesn't go in much more detail than that besides conditions like fog potentially worsening detection ranges.
Usually combined with the IR sensor as per the summary sheet, it can detect "cold" targets but is also not ideal in poor environmental conditions and mostly useless at night.
There's not much theory behind this, you should just understand what these things are but you do not need to know how they work in Arma or real life, as you'll just realistically have to adapt to whatever situation you find yourself in (e.g. if you can't lock onto a target from further away, you'll just have to get closer or improvise).
Top tip: Vehicles can send information between each other via Data Link, essentially allowing for improved awareness for all connected vehicles and easier targeting.