The blue kind is a mechanical relay. They’re decently robust but 10A might not be rated high enough for your vacuum, depending on where you are in the world. The opto-coupler isolation is also a bit of a gimmick.
There are 2 ways to wire this but the simplest one would be to connect DC+ to pin 2 or 4 of TB2 (+Vmot) and then connect both IN and DC- to pin 1 or 3 (the switched pin). If it doesn’t turn on when the output is switched, you’ll need to change the jumper on the relay board so that it switches on when IN is low.
The other way to wire it would be to provide +24V and 0V to DC+ and DC- respectively then to connect pin 1 or 3 of TB2 to IN. You may need to also add a ~1k to 4.7k resistor between DC+ and IN to ensure that it turns on in this case.
The white one is a solid state relay which will be simpler to wire up but may need heatsinking depending on the load. You’d hook + to pin 2 or 4 of TB2 and then - to pin 1 or 3 of TB2. With 10A going through the it’ll get a bit warm. If the backplate gets too hot to touch for more than a second or two with your finger, it might need to be bolted to something metal or a small heatsink to help spread the heat out.