Do they use a tungsten wires or something else?
Falcon 9 uses TEA-TEB to ignite its engines both on launch and during all the landing burns. The mixture is pyrophoric - it ignites spontaneously when it comes to contact with oxygen (both in air or the liquid supercooled).
The rocket carries limited amount of the stuff so the number of possible re-ignitions is limited too.
Currently it needs to re-ignite 3 engines for the boostback burn (when doing RTLS or limited downrange landing - usually for LEO missions, GTO ones do not execute this burn), 3 again for the re-entry burn and 1 or 3 for the landing. That means it should have enough for 7-9 uses.
The second stage may need to reignite its engines too, depending on the mission so it needs to carry some amount of the igniter fluids too.
On the first launch of Falcon Heavy there was some problem on the center core which caused the outer two engines to run out of igniter fluid and not start for the landing burn, meaning the core was not able to slow down enough and hit the water at about 500 km/h (~300 mph). The cause is currently unknown or at least unpublished - possibilities including bad valve, human error or some FH specifics.
From what I gathered reading various comments about that incident, it seems that there is no specific "dosage" of the TEA-TEB mixture per engine startup, instead the mixture is allowed to flow to the engine until proper startup is detected. But I do not know about any official source for this and it may be just wrong understanding on my side.
Another speculation on my part: because SpaceX uses 1-3 or 1-3-1 scheme for multi-engine landing burns and according to currently public information both outer engines failed to ignite, it seems reasonable that there is just one common reservoir of the TEA-TEB mixture per stage, supplying all 3 restartable engines, instead of a reservoir per engine. That way the center engine would start fine because it was first to get started during the burn, but some problem (possibly greater airflow because of heavier and thus faster center core?) made it use more fluids for the last ignition, leaving just scraps for the outer engines.