Paraffin wax: Paraffin wax is one of the most common candle waxes. That's why most candles you buy off the shelf are usually made with it. Paraffin can be really easy for companies to produce since it's made from refining crude oil and petroleum. AKA the candles are made from the leftover parts that probably would have been thrown out. It can come in two different melting points, that mean either you'll buy soft wax, or harder wax.
The pros for using paraffin are that most candles sold in stores are made from it so you'll have lots of buying options, if you're interested in DIY, paraffin candle making guides are more common than natural wax guides since paraffin is so easy to find. And it usually holds fragrances and dyes really well since it can have lots of additives added to it, so it's great for craft projects. The cons, would be that since the paraffin was made from crude oil or worse, it can cause allergic responses in sensitive people. Or if you really care about the environment then paraffin probably wouldn't be a good pick.
Gel wax: This type of wax isn't technically wax. It's a modern form, made from mineral oil and/or synthetic polymers. A huge pro of using this gel is that it can make really cool candles! It can be made into candles that have floating objects in them or different scenes, like ocean themed ones, because the gel is usually transparent. Gel waxes don't have regular melting points. they can just become liquid at certain temperatures. But there are softer or harder polymer based gels that you can buy. Another pro is that these gels can burn more slowly than other candles.
But a warning is that these gels need to be in heat resistant containers, like especially heat resistant glass. Otherwise the container could explode.
Stearin: I thought this wax sounded so cool since it's made from animal fat! Or at least it's made from a blend of animal fats and other waxes. It can be added to other candle waxes like paraffin, to make them more usable for candle making. Like making the candles burn longer or making more stable candles. So generally it's not supposed to be used as a candle wax by itself, it's just to help bring up the quality of other candle waxes. Or also to add glossiness and make the colors more vibrant. But one warning is that you're never supposed to use it in a rubber mold, or it will melt the rubber!