A cartel is a group of firms acting together to restrict competition. A group of producers agree not to sell below a given price, or to restrict output (which will increase prices) or not compete on each others territory, etc. By acting together the firms in a cartel are able to give themselves the same, or similar, pricing power to a monopoly.
In general, cartels are illegal in any economy that makes any attempt to be a free market. They are precisely what competition laws (called anti-trust laws in the US) are designed to prevent. However there are, and have been, various exemptions for particular industries. Some cartels in international markets are operated by national governments; the best example of this the oil cartel OPEC. OPEC sets output quotas rather than selling prices, but as price depends on supply the effect is the same.
It is difficult to enforce the competition laws designed to prevent cartels from controlling prices. One of the problems is actually proving collusion. Any discussion between the operators of an illegal cartel is likely to be carried out covertly. They may sell at similar prices, purely as a result of the operation of a normal competitive market. Even suspiciously high margins may be explained away by an oligopolistic market — and it is in oligopolistic markets that cartels are most likely as a small number of large competitors can cooperate covertly.
There are also many things that companies within a particular industry may do, that are capable of legitimate explanation, but which may have the effect of setting minimum prices or of locking out other competitors or of erecting barriers to entry. Any cooperation between companies within an industry should be regarded with suspicion by consumers (and, hopefully, regulators).
Cartels are sometimes ended by competition regulators. They may also collapse because a member of a cartel sees an opportunity to gain an advantage by breaking whatever agreements bind the cartel. For example, by suddenly cutting prices a company may be able to capture enough market share to more than compensate for the lower margins that result.