This achievement is for killing 50 Necromorphs with the flamethrower. The flamethrower schematic can be found in Chapter 6 in a classroom on your right (previously locked, I think it was the room with the first exploding baby and the teacher) after leaving the gymnasium after the ambush there.
I found the easiest way to get this was killing infants (not the exploding babies though) and parasites as the Flamethrower is fairly short range but useful against swarms. Cysts DON'T count towards the achievement. Use primary fire rather than secondary as secondary is not especially effective for the amount of ammo it uses.
It can also be effective in one on one situations with larger enemies eg Slashers as flaming them tends to halt their movement towards you.
There is a counter in the "achievements" menu that you can access by hitting start. Note that progress towards the achievement is recorded at checkpoints so hitting "checkpoint restart" will wipe any progress since the last checkpoint, so you can't just find a good spot, spam it, and then restart the checkpoint.
I haven't yet tested whether saving a game, killing enemies, making it to a checkpoint, then reloading the save and repeating the section would work. Edit: NorrisMM has suggested this will not work either (i.e. you need to do it the hard way!).
Edit: If you're going to downvote, at least explain why.