negative versus positive control
Think of two ways to manage a faucet. One way is a clamp on the hose: by default water could flow, and you stop it by squeezing the clamp shut — that is control by blocking. The other way is a hand on the tap: by default nothing flows, and water comes only when a hand turns it on — that is control by helping. Bacteria regulate genes in both of these styles, and the names simply describe which way the controlling protein works.
In negative control, the regulatory protein is a repressor: when it binds the DNA, it blocks transcription, so the gene's default tendency is to be on and the regulator's job is to switch it off. In positive control, the regulatory protein is an activator: when it binds, it helps RNA polymerase get started, so the gene's default tendency is to be off and the regulator's job is to switch it on. The two are not opposites you must choose between — a single operon can use both at once. The lac operon is the classic example: a repressor provides negative control (off unless lactose is around) while an activator called CAP provides positive control (sluggish unless glucose is scarce), and the gene fires strongly only when both signals agree.
Knowing whether a control is negative or positive tells you what happens when you delete the regulator, which is exactly how biologists worked these systems out. Delete a repressor and the gene becomes stuck on (it was being held off); delete an activator and the gene becomes stuck off (it needed the helper). This either-blocks-or-helps distinction is one of the most useful first questions to ask about any gene switch, in bacteria and beyond.
A lacI mutant that cannot make functional repressor leaves the lac operon switched on even with no lactose around — telltale of negative control, where removing the regulator releases the brake. A broken activator would do the opposite, leaving a gene stuck off.
Negative control blocks (default on); positive control helps (default off). Many operons use both.
Negative versus positive describes what the regulator does (block or help), and is independent of whether a system is inducible or repressible. A single operon often combines both kinds of control, so the labels are not mutually exclusive.