In German most countries are, as in English, just called by their names, such as Deutschland, Frankreich, and Belgien; but some country names require the definite article, just as in English: the United States or the Netherlands. The rules for whether a country name requires a definite article or not are not always the same as their English equivalents, however, and have a specific feminine or masculine gender too:
Für mich war es schon ein Erfolg damals mit in die Türkei ins Trainingslager zu kommen.
For me, it was already a success to come to training camp in Turkey at the time.
Caption 17, Fußball: Spielerportrait Lars Stindl
To "come to training camp in the Turkey" would certainly purvey a different meaning than intended, but in German, the country Turkey requires the feminine definite article: die Türkei. Some other countries with the feminine definite article are: die Mongolei, die Schweiz, die Slowakei, and die Ukraine. Furthermore:
Es gibt maskuline Länder, zum Beispiel „der Iran".
There are masculine countries, for example "the Iran."
Caption 55, Deutschkurs in Tübingen: Präpositionen
Obviously "the" is used before "Iran" in the English translation only to emphasize the use of the definite article. Some other countries requiring the masculine definite article are: der Irak, der Jemen, der Kongo, der Libanon, der Sudan, and der Tschad. The only countries that require the neuter definite article are those that use the word "Kingdom" in their name such as das Vereinigte Königreich (the United Kingdom), but this is clear in the grammar, since das Königreich is a neuter noun.
Last but not least come the countries that require the plural definite article:
Frankreich war weitaus rückständiger als die Vereinigten Staaten.
France was much more backward than the United States.
Caption 45, Malerei: Impressionistinnen
In most cases, the countries that use plural definite articles are the same as those that do so in English: die Bahamas, die Niederlande, die Philippinen, die Salomonen, and die Seychellen.
A quick word of warning regarding the use of definite articles and country names: If a country with a non-plural definite article is preceded by an adjective, then the definite article is referring to the neuter noun das Land (the country, the nation) and always requires the neuter definite article. Even countries that do not require the definite article in normal usage get the neuter article das if they are being described preceded by an adjective. This is actually easier in practice than theory: Das schöne Frankreich, das teure Norwegen, das warme Brasilien.
Browse through Yabla videos and find some country names being used in context and in different cases like dative, accusative, and genitive. For an interesting in-depth article on the topic, see the ever-fascinating Zwiebelfisch-ABC series from Der Spiegel.