The word country probably derives from the Latin word "contra" which means to set to against. A contra in accounting for instance, is something which is set against another entry; usually to remove it.
A country then, is probably something which is set up against other countries as distinct from them. This usually implies a sense of centralized government and political geography but even here we run into our first problem.
What is England? The metonym has at times referred to the whole of the United Kingdom and even for the British Empire but England is a thing without its own centralized government. Before 1701 with the Act of Union with Scotland which created the United Kingdom, England had its own government at Westminster but since then it has had to share.
Scotland, Wales and Northern Ireland all have their own distinct devolved parliaments and later on this year, Scotland goes to the polls to decide whether it wants to dissolve the Union but England of itself has no distinct parliament. Is England therefore a country?
Somaliland has its own parliament, its own constitution, political contacts with ten other countries and issues its own currency, the Somaliland Shilling. Nevertheless, Somaliland is not officially recognised by the UN; although it issues its own passports, they are not recognised by any other country in the world, though some countries accept them as accompanying travel documents.
If England has no parliament of its own and is a country, then why is Somaliland which has its own parliament not a country?
The People's Republic of China refuses to accept the claim that the Republic of China (the official name for Taiwan) is a country; likewise the Republic of China (Taiwan) accept the claim that The People's Republic of China is a country. Thanks to this official hostility, only 21 countries maintain official diplomatic relations with Taiwan and at international events like the Olympics and FIFA football tournaments, the country is referred to as Chinese Taipei and does not fly its official flag.
Taiwan was a founding member of the United Nations but was officially expelled by United Nations General Assembly Resolution 2758 which replaced its membership with that of The People's Republic of China.
Partly because of The People's Republic of China's permanent place on the UN Security Council and their "One-China" policy, all attempts for Taiwan to apply for membership to the UN are pointless as China would simply veto it.
If Taiwan has a distinct parliament, trades with other countries but lack official recognition at the UN, is Taiwan a country?
Taiwan is probably the easiest case to lay out here, though Somaliland is kind of a good candidate for something as close to a country if it isn't one.
Then there's Nagorno-Karabakh, Abkhazia and Transnistria which are recognised by countries which aren't countries. Places like Kosovo, Palestine and South Ossetia, which a which are recognised by some countries which actually are countries and really weird disputes like Israel which isn't recognised as a country by 32 other nations, the two Koreas which refuse to recognise each other and Armenia which has had a long running petty dispute with Pakistan.
Really as far as I can make out, the only real test whether a country is a country or not is whether other countries think that that country is a country or not; parliaments help though.