der Tscheche
Czech man
A male person who comes from the Czech Republic or has Czech citizenship.
Mein neuer Kollege ist Tscheche und kommt aus Prag.
My new colleague is Czech and comes from Prague.

Czech
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Czech man
A male person who comes from the Czech Republic or has Czech citizenship.
Mein neuer Kollege ist Tscheche und kommt aus Prag.
My new colleague is Czech and comes from Prague.

'Der Tscheche' (the Czech man) is a noun of the N-declension. This means it takes the ending '-n' in all cases except the nominative singular. This is an important rule for many nationality terms ending in '-e'.
Ich habe gestern mit einem Tschechen gesprochen.
I spoke with a Czech man yesterday.
The word belongs to a family: the country is called 'Tschechien' (Czechia/the Czech Republic), the female person 'die Tschechin' (the Czech woman), and the adjective 'tschechisch' (Czech).
Die tschechische Küche ist für ihre Knödel bekannt.
Czech cuisine is known for its dumplings.
Note the difference: 'Der Tscheche' (the Czech man) is a noun and is capitalized. The related adjective 'tschechisch' (Czech) is written in lowercase.
Er ist Tscheche, spricht aber perfekt Deutsch.
He is Czech, but speaks perfect German.
Slovak
'der Tscheche' and 'der Slowake' both refer to a male person by origin or citizenship. The difference is the country: a 'Tscheche' comes from the Czech Republic or has Czech citizenship. A 'Slowake' comes from Slovakia or has Slovak citizenship. Learners may confuse the words because the countries are neighboring countries and used to be part of Czechoslovakia together. So the important point is: the Czech Republic and Slovakia are two different countries today.

Der neue Kollege ist Tscheche und kommt aus Brünn.
The new colleague is Czech and comes from Brno.
Beim Sprachkurs sitzt ein Slowake neben mir.
A Slovak man sits next to me in the language course.