They are not a country but rather a handy way of describing two entirely separate Crown Dependencies, which are each self-governing in all respects except for Defence and Foreign Affairs, which remain the responsibility of the United Kingdom.
The Islands fall into two separate Bailiwicks (historic feudal divisions), each of which has its own separate government. Guernsey, Alderney and Sark (comprising the Bailiwick of Guernsey) is effectively a Customs Union with no customs controls between them (despite the fact that Sark levies taxes on alcohol and tobacco at a much reduced rate to the rest of the Bailiwick!)
The Channel Islands have been inhabited for over 5,000 years and have a long and colourful history. During WWII they were occupied and retain many military structures, both from this period and from the time of the Napoleonic Wars.
They count their independence of any ties to France from the year 1204.
Today, the Islands' Head of State is the Queen of United Kingdom who is represented in the Islands by her Lieutenant-Governors. Her role derives from Her status as the successor to the now-defunct Dukedom of Normandy (the Islanders' version of the Loyal Toast, is "The Queen, our Duke". The Islands laws are a mixture of local legislation, customary law (heavily influenced by the English Common Law), Acts of the UK Parliament which have been extended to the Islands and (some) European Union Law in respect of (e.g.) the free movement of people and of goods. The Islands have their own tax systems, currencies (at par with the GBP), banknotes, and individual Parliaments. The relationship with the EU is complex and little understood (they are in the European Union Customs Union but outside the ambit of fiscal and social legislation, for example).
English is spoken throughout the islands, but there are still remnants of the old Norman patois.
Like the UK, the Channel Islands are outside the Schengen Agreement but form a Common Travel Area with the UK, Republic of Ireland and Isle of Man.
Due to their quasi independent status, taxes on drink are lower than in the United Kingdom, and so prices can be lower.
A lot of the people here are friendly and will be happy to assist you around the islands.