inducible versus repressible system
Some lights in a house come on only when they are needed — a closet light that turns on when you open the door. Other lights stay on all the time but switch off when something tells them to — a porch light on a sensor that goes dark once the sun comes up. Bacterial operons split into these same two habits, and the words simply say which habit a given operon has.
An inducible system is normally off and gets switched on (induced) when a particular small molecule appears. These are typically operons for breaking down a nutrient: there is no point making the digesting enzymes until that nutrient is present, so the food itself (or a relative of it) acts as the inducer that flips the switch on. The lac operon is the model inducible system — off until lactose shows up. A repressible system is the mirror image: normally on, and switched off (repressed) when a particular small molecule builds up. These are typically operons for building something: the cell keeps making the building enzymes until it has enough of the product, at which point the surplus product (the corepressor) tells the operon to shut down. The trp operon, which makes the amino acid tryptophan, is the model repressible system — on until tryptophan accumulates.
The two ideas capture a sensible economy: switch a breakdown pathway on when its raw material arrives, switch a building pathway off when its product is plentiful. It is worth keeping this distinction separate from negative-versus-positive control: inducible/repressible says whether the small molecule's presence turns the gene on or off, while negative/positive says whether the regulator works by blocking or by helping. A system can be inducible and negatively controlled (lac), or repressible and negatively controlled (trp), and other combinations exist too.
Lactose induces the lac operon: present the sugar and the cell turns the digesting genes on. Tryptophan represses the trp operon: let the amino acid build up and the cell turns the building genes off. One reacts to a food, the other to a finished product.
Inducible: off until a nutrient appears. Repressible: on until a product piles up.
Inducible/repressible (does the small molecule turn the gene on or off) is a different axis from negative/positive (does the regulator block or help). A system has one label on each axis, so do not treat 'inducible' and 'negatively controlled' as alternatives.