I have a few confusions about this board:
1- What type of capacitors are these?
2- When should I use them and why are they being used here? There are ceramic caps on the board as well, so the manufacturer has deliberately decided to use this type of capacitor for certain parts instead.
3- I'm assuming that capacitor in the middle next to the big IC is a decoupling capacitor. I've always heard that you use 100nF ceramic caps for decoupling but this board is using this type of capacitor. Why?
Probably used here for supply decoupling, but it's hard to be certain.
You could use them for that, to block DC on a signal line, as part of a timed RC circuit, etc. etc. They're just capacitors like any other type, basically.
Sometimes there are somewhat more arcane reasons why people use certain types of capacitors, such as in high-end audio equipment. It's a can of worms I'd rather not open here, but trust me, capacitors are a THING in the audio community.
Whether you should use these...I don't know, it's up to you. Personally I usually just use ceramic SMD capacitors if small values are OK and electrolytics if they need to be bigger. I don't really see any reason to make things more complicated. Yes, there are several types of capacitors, each with their drawbacks and advantages, but for 99% of the applications we see here on this forum, the type doesn't matter all that much. The main distinction you generally need to know is if a cap is unipolar or bipolar, i.e. is it DC only or will it work for AC (to put it a bit simplified).
I would beg to differ. On that relay board I can see Tantalum capacitors, the orange tiny things amongst the resistors. Aluminium electrolytic , polyester and ceramic. It depends on what the capacitor is being used for. There is stability and tolerance to think of, a ceramic capacitor is nether temperature stable, voltage stable nor has a very precise value. So they only any good for supply decoupling, and that is only one use of a capacitor.