When a spoke is broken the whole wheel is off balance. Best case: replace the spoke, some fine tuning, back on the road. A tensiometer helps to maintain equal spoke tension when building and truing a wheel. It is not strictly necessary.
A damaged spoke can be found by visual inspection (chain drop on the rear wheel). Most spokes fail due to metal fatigue, ultimately caused by poor build quality of the wheel. Too low spoke tension, no "setting" (bending the elbow of outside spokes a bit), now stress relief and messed up torsion are the main factors there.
When more than two spokes break on a wheel: replace all of them. Build quality was shite (most likely).