I'm currently studying formal languages. My lecture states that, while the words of a language are finite, the sentences build with the underlying grammar are not. But I don't get why this should be generally true. I can imagine grammar which leaves me with a finite amount of sentences.I agree that one could create a language with infinite amount of sentences but I can't see why it should be generally the case. 
Why is the quantity of sentences not finite?
formal languages
The statement is not true for arbitrary languages and grammars. For example, the grammar $G$ consisting of the rules$$S \to a|b|C\\C \to cD\\D \to d|\epsilon$$can produce the words $a$, $b$, $cd$ and $c$, so $|L(G)| = 4$, therefore $L(G)$ is finite.The alphabet (words of the language) is finite by definition. The language may or may not be finite. A trivial example of a grammar producing an infinite language is $G' = (\{ S \}, \{ a \}, P, S)$ with $P = \left\{ S \to a|aS \right\}$. $G'$ produces the infinite language $L(G') = \{ a^n : n > 0 \}$.Both finite and infinite languages exist and can be generated by grammars.Note that all finite languages can be expressed through regular grammars and are therefore regular languages. However, not all regular languages are finite (see $G'$) The class of regular languages (type 3) is a subset of the class of context-free languages (type 2) which is a subset of the class of context-sensitive languages (type 1) which is a subset of the class of recursively enumerable languages (type 0):$$\text{Finite languages} \subset \text{Type 3} \subset \text{Type 2} \subset \text{Type 1} \subset \text{Type 0}$$(And these are only the languages you can generate using formal grammars, in fact, recursively enumerable languages (type 0) are just a subset of the set of all languages.)