I fully understand why you would wonder about that as the Spanish have truly invented their own way for preparing coffee and their coffee drinks have specific names.
So knowing what their different coffee drinks are will definitely help you order something you like.
Spain is actually one of the largest consumers and producers of coffee worldwide. It has a long tradition of drinking coffee since the 1780s, when it became popular among well-off people in Spain.
Spanish coffee culture is known for serving espresso in small cups or glasses, often with sugar. Cafés are important to Spanish culture; they are well-known meeting places, where people go not only to drink coffee and talk, but also to read newspapers, eat breakfast, play chess or dominos etc.
Many a time the Spanish serve the coffee in a glass so if you don’t like that you better ask for a cup (“tasa”) while you order. Also, in some parts of Spain other than the big cities, there are not many dairy-free options to choose from in a coffee shop. Your best bet would be soya milk.
As for other milks, if you don’t want full fat you can choose between skimmed (“leche desnatada”) and lactose free (“sin lactosa”).
And here are the most common coffee types you will find in a Spanish coffee shop:
Café Solo: a single shot espresso
Café doble: a double shot espresso
Americano: café solo with hot water
Cortado: a shot of espresso, with a splash of milk
Largo: a double shot of espresso, with a splash of milk.
Café con leche: half espresso and half milk
Leche Manchada: a glass of milk with a drop of coffee in it
Carajillo: a café solo with a shot of dark rum or whiskey
Descafeinado: a decaf option
Café Bonbon: a café con leche with a splash of condensed milk
Based on the above you should be able to find a type of coffee that best suits your taste. Mind you, you can also find third wave coffee shops in many big cities in Spain, where they serve more well-known coffee drinks like lattes, flat whites and cappuccinos.
Let me know if you have any more questions regarding Spanish-style coffee!