how can we measure a resistor's resistance when its fixed on a board in a circuit?

 

Some can be measured in-circuit; others, you will have to unsolder one end.

 

You can measure it in circuit just as you would out-of-circuit, one multimeter probe on each end. But the measurement won't mean much if you don't know what might be in parallel with it.

One useful and definitive result though, is when a resistor has failed open like a fuse, and shows "infinite" resistance on the meter.

 

One useful and definitive result though, is when a resistor has failed open like a fuse, and shows "infinite" resistance on the meter.

Will the parallel resistance from the rest of the circuit not still be there when a resistor blows apart?

It is this parallel resistance the get in the way of any in-circuit measurement.

Sometimes you can't easily remove the resistor, or even one end so you have to play Sherlock Holmes and try to find the voltage across it and the current through it. You don't have to power the circuit from its own supply for this, in fact a lower supply is better and some ohmmeter ranges use a low voltage that does not switch on semiconductor junctions for instance, You can sometimes deduce the current by measuring the voltage a cross a resistor that is in series with the suspect one.