The brass bit you can see is called the insert.....one is for water on/off the other is the temperature control - which looks thermostatic.
You'll need the make of the tap / shower first and inserts are available from showerdoc or similar who can supply quickly and they have an excellent illustrated website. The inserts are very expensive for what they are so you may consider a complete bar replacement.
If you can isolate the water supply to your shower, so much the better!
Changing that whole bath/shower thermo tap isn't too hard if you have the correct size of adjustable spanner, are reasonably handy and can shut off your hot and cold water.
You can pick up a cheapo thermo shower filler tap from fleebay, shut off the water, and then loosen the 2 nuts, get rid of any PTFE tape from the threads (big no no) and then fit the new one, pipe gaps should be the same (150mm).
Looking at the pics again I just realised that you have Z/S connectors to adjust the gap from the probable 180mm pipes coming out of the wall, to the 150mm gaps that the shower filler needs. So the PTFE tape will be needed at the wall connections but not at the filler connections, they should have rubber sealing washers in them.