A state of emergency is an official declaration by the government that the situation is a lot bigger than they can handle. It gives them additional powers, usually suspending many freedoms people normally enjoy (though it’s not always clear why these are suspended in the case of an actual emergency). It also allows the government to commandeer and ration services like power distribution, water supply etc. It can even impose travel restrictions and curfews.
Normally, there are conditions and limitations to the state of emergency (though these can be circumvented by certain governments). It’s not unusual for there to be an ongoing state of emergency.
In the UK, only the Sovereign on the advice of Privy Council or a Minister of the Crown can introduce emergency regulations during a state of emergency. These regulations can be revoked within 30 days, but can be extended for up to 90 days. During this time basic human rights can be suspended, but the parliament is still able to function.
The reason why a state of emergency is declared usually comes down to the fact that a situation is bigger than the local governments can deal with on their own. The state can then send in additional resources to help deal with the problem. For example, if there is a blizzard that dumps 3′ of snow on the area, it would likely be declared a state of emergency which would then lead to extra plow crews being sent in to clear roads and provide assistance.