How did countries get their English names?

Depends.

Recent country names are carried across from whatever the country is calling itself, without much alteration: Bhutan, Nepal, Senegal, Angola.

Neighbouring countries that England had close contact with traditionally would have the most diverse names—mainly based on what those countries called themselves, but looking Germanic, and not made to be consistent. Scotland, Ireland, Denmark, Wales, France, Flanders, Norway, Sweden.

Otherwise, the main source of country names for the “Old” World—Europe, North Africa, major countries of Asia—is Latin, and indirectly Greek, as the prestige languages of English learning for a very long time. Thence the –ia suffix, which actually goes through three iterations:

• –ia for more recent loans, straight out of Latin, and less familiar countries (Albania, Persia, India, Slovakia)
• –y for Middle English and Early Modern English (from French), many instances of which later went to –ia (Turkey, Hungary, Italy; Normandy, Picardy; Indies for India, Candy for Candia = Crete)
• –e or nothing for very old and familiar country names (France, Spain)

Of course, see also List of country name etymologies on Wikipedia.