What is overboost?

tl;dr: Overboost is too much air, usually at too high a temperature. Detonation is likely to follow.

Remember, the engine is just an air pump. All the turbo is doing is making it easier for air molecules to get into the intake side of the engine. Of course, the turbo has finite efficiency: it is pushing in more air molecules but it's also increasing their net energy (i.e., intake air temperature goes up).

The engine needs to add enough fuel molecules to combust with those air molecules. Because of the increase in air temperature, it sometimes needs to add even more fuel as a coolant (which seems wasteful because it is). The engine's goal is to make sure that there's just the right air-fuel mixture at just the right temperature to ignite exactly when it's supposed to.

Too much boost (overboost) can lead to two big problems (well, lots of possible problems but let's stay at a fundamental level for now):

Lean is bad: If you have too little fuel for too much air, you have a lean condition. In this situation, the fuel-air mixture is actually easier to ignite with increasing temperatures (e.g., under boost). In fact, it can sometimes suddenly ignite due to hot spots in the cylinder or just the fact that the whole mixture is above its activation energy.

Quoting Corky Bell of Maximum Boost fame:

Any misfires at full throttle induced by a lean condition are serious and must be dealt with prior to operating at that boost level again. A lack of fuel raises chamber temperatures dramatically. Heat is the cause of detonation, which is the nemesis of high performance.

The consequence is that the mixture in the cylinder ignites at the wrong time. This cause all kinds of bad mechanical problems, sounds awful, can break your engine and definitely doesn't make you feel speedy.

Terminal detonation is exactly what it sounds like: critical pieces of the engine itself explode away.

Too much energy is bad: It's possible to have too much energy in the system even when things are going "perfectly." Imagine a theoretical system where your turbo is 100% efficient (adds no heat during compression), your fueling system can pump in arbitrary amounts of fuel and your engine computer can keep up with all the demands. Even in that highly imaginary situation, if you put enough energy in the cylinder, it's going to break. The fundamental structural components of the system will eventually fail given enough stress.

In real life, such a perfect system is rare to impossible given our finite budgets. However, ...

apparently there is a good type where the engine produces more power for 5 seconds or so. Why just for say 5 seconds and not all the time?

The answer is math (specifically probability). This sort of feature is most useful in auto racing where the driver needs that little extra edge to make the critical pass. They're trading a burst of energy for an increase in system temperature, net strain on the engine and briefly increased risk of total system failure. Roll the dice and hope it doesn't come up snake eyes during that five seconds.

In a race, that's often a reasonable trade-off. Obviously, those of us driving on the street are usually operating without the benefit of an auto-racing team's budget.

Funnily enough, overboost is usually straightforward to correct. It's almost certainly a wastegate problem:

A wastegate is a valve that diverts exhaust gases away from the turbine wheel in a turbocharged engine system.

By diverting exhaust gases away, the engine computer can hold the turbine to a peak speed. As a consequence, the compressor side will also be limited to a peak boost. If something happens to keep the wastegate from functioning as needed, your turbo is going to try to keep accelerating up to infinity.

As we have discussed, it isn't going to get there. Boom!