A tungsten jig is a jig head poured from tungsten instead of lead. Tungsten is denser, so for the same size head it weighs more, sinks faster, and reads bottom contact harder than a lead head of the same profile - you feel every rock and ledge through the line. Or, for a given weight, you get a smaller, more compact head. Pair it with a small soft-plastic body (a Ned-style stick, grub, or worm) and you have a compact, sinking finesse rig.
Sizes 1-4 cover most trout work. Stick to natural, subdued colors - black, brown, and olive - which match crayfish, sculpins, and nymphs along the bottom. In clear water the muted tones get more eats than anything bright.
This lure works in cold water, roughly 40-60F, when fish hold deep and slow. It wants normal flow and clear-to-stained water - enough visibility for a fish to track a slow-moving target. Fish it slow: cast out, let it sink to the bottom, then drag and hop it back with long pauses. Most strikes come on the fall or during a dead pause, so watch your line and stay in contact.