Autarky is economic independence or self sufficiency, and should not be confused with autarchy, a separate political concept. Historically, some nations have exhibited autarky, but today, it is extremely rare. There are a number of disadvantages to an entirely closed economy which force most nations to abandon attempts at autarky. There are also, of course, some distinct advantages.
The idea behind autarky is that when an economy is self sufficient, it can theoretically be stronger. When a nation can provide all of the goods and services it needs internally, it is not forced to rely on other nations. This can be a political strength for a nation which does not want to make concessions in order to establish or maintain trade relations. If a country has a closed economy, for example, economic sanctions from other governments which are intended to force that country to address something will have no effect.
Another reason for a closed economy to be appealing is that it can allow a country to control the supply of technology, goods, and services. For example, development of weapons systems may be done in an entirely closed economy for the purpose of keeping such systems out of the hands of other nations. A country with a supply of a rare resource can use autarky for political clout by keeping supplies of that resource out of other countries.
One major drawback to autarky, however, is that very few nations can actually accomplish it. Trade with other countries allows nations to access products and services they cannot produce on their own. For example, a country without a supply of timber needs to be able to get wood from somewhere. In nations which have tested autarky as an economic system, citizens can experience privation, and often black market activity gets intense as citizens attempt to access goods from outside the country clandestinely.
Other issues with a supposedly self-sufficient economy can include diseconomies of scale which are caused by limitations on the supply of raw materials. In this case, per unit costs of production go up as a result of economic limitations. It can also be hard for a nation to develop new technology, new services, and new approaches to doing business when the economy is isolated and people cannot exchange products and ideas with the outside.
Cases in which autarky have been effective have primarily involved nations with numerous colonies. In these instances, a nation can meet its needs from within its borders and the borders of the colonies it controls.