A component of a board from an appliance has failed and I would like your help identifying it, in order to replace it.
Below is a picture from the board, the arrow points to the failed component:
I'm not convinced that the surface-mount device (resistor?) pointed at with your red arrow, has necessarily failed. Yes, something in that approximate area has got hot, but that doesn't mean it was this SMD.
If you reverse-engineer that part of the PCB and draw the schematic (assuming you can't get a schematic any other way), this would help you to understand more. For example, I've marked the location of a through-hole component with a light-blue line on the edited version of your photo below. That component seems to be in parallel with the unidentified SMD component. That would be odd, if both components were resistors, so at least one of those two parallel components is probably something else.
The PCB may have got hot in the area of that through-hole component, rather than the SMD component next to it. If I was trying to diagnose that PCB, I would be drawing a schematic of the part of the PCB, to help with understanding the design.
The main reason I wanted to write this answer, was to highlight this point: It seems that at least one relevant solder joint is "dry", perhaps actually now loose (fractured) and therefore either high-resistance and/or intermittent. I've marked it with a solid green arrow on the edited copy of your photo above.
That "crescent" of darkness you can see in the solder joint, is classic of that type of problem. Notice how the suspect solder joint is very "crystalline-looking" - compare that with the shiny solder joint below it. Many repair engineers would reflow all suspect solder joints on a board like that.
Therefore it's possible that:
no component has actually failed;
the suspect solder joint is the actual cause of the problem;
the PCB has got hot due to high current going through the higher-than-normal resistance of that suspect solder joint, to whatever is connected to the black wires on the right of your photo in the question (I'll guess perhaps some kind of motor?).
Obviously component failures are also a possibility. However no component values can be identified from the photo you supplied.
Someone in front of that actual PCB would need to investigate further, to see which solder joints are faulty, and which components (if any) have failed. Since this is a mains-powered device, please be very careful if you decide (at your own risk) to do any troubleshooting - both because of the potentially-lethal voltages, and also in case something mechanical (e.g. a motor) suddenly and unexpectedly moves, while your hands or face etc. are close to it. If in doubt, contact someone local to you who is qualified / experienced for this type of work.
It doesn't look failed to me. It's a thin film resistor with no top. There no way to determine if it should have a black epoxy top or not without seeing a part number, working unit or schematic.
If it did fail, then it's likely that the value has changed, finding another unit will be necessary to determine the value.
Usually when resistors fail, they also show signs of heat damage on the part (which is not apparent in the pic above) and\or on the PCB below. There are also PCB trace fuses on the PCB and they are intact.