Actually IIRC a nuclear reactor going critical means that it started working. The Bad Thing is it going prompt-critical. Then again, I don't know shit, so I need someone to whale on me here.
Quick summary;
A nuclear reactor works by being a controlled chain reaction. Each atom that decays emits a number of neutrons. Each neutron has the potential to cause another atom to decay. You have more than one neutron being emitted by each decay (on average, the exact number can vary depending on the decay you are looking at), but at the same time actual interaction between emitted neutrons and other atoms is rare.
Reactors use a combination of
neutron moderators (slow neutrons making them more likely to interact and trigger another decay) and neutron absorbers (do what it says on the tin) to keep the ratio of neutrons emitted to neutrons that trigger another decay at 1:1. This ratio is either described as the
effective neutron multiplication factor (denoted as k) as a straight up ratio or as the reactivity. Reactivity is just k-1.
When the reactor is critical k = 1 (reactivity = 0) so each decay triggers exactly one other decay and the energy output remains constant across time. A reactor with k < 1 (negative reactivity) is sub-critical, and experiencing a reduction in energy output over time. k > 1 (positive reactivity) is supercritical and increasing in energy output.
Supercritical reactors can be dangerous, but not usually. It just means the energy output is being ramped up and is only dangerous if there are positive-feedback loops and a lack of safety measures to stop it running away entirely. Most power plants are built with some degree of negative feedback to counter such a runaway event.
Prompt-criticality is a special case of supercriticality. Basically the neutrons emitted in a nuclear decay are split into prompt (emitted immediately) and delayed (emitted seconds to minutes later). The delayed neutrons only make up ~1% of all emitted neutrons, but are useful as they effectively slow down the rate of increase of power in a supercritical reactor. Most reactors are designed to be critical only when taking into account the delayed neutrons; you still have k=1, but some of the triggered reactions happen much later than the triggering event. This slows things down enough that mechanical systems (control rods mostly) can respond and control the energy changes. A prompt-critical reactor is one where the reactor is critical through prompt neutrons alone. Now any increases in power are going to be immediate, far faster than mechanical systems can respond. This is obviously dangerous, although doesn't always mean a complete disaster.
Generally you want to keep a reactor in a delayed-critical state at all times to retain control over it. So when ramping up energy you go super-critical, but remaining in the window where it's only super-critical due to delayed neutrons. That is what makes nuclear reactors relatively slow to ramp up. You could theoretically design a reactor that uses prompt-criticality to get up to it's operating level in extremely short time, but that's going to be hard and dangerous. I think I remember reading about someone who had tried it in a simulator and in models, but strongly doubt it would ever actually be used in real life.
NB: Fast reactors don't use moderators and instead use decays that can be triggered by fast neutrons. Most notably U-238 can be transmuted into Pu-239 in this way, which is the breeding in fast breeder reactors. These events do occur in normal nuclear power plants as well, but to a much reduced degree.