Learn Icelandic: Ég er þyrstur og vil vatnið. - I am thirsty and want the water.

In Icelandic, adjectives that describe a person must match that person's gender. Þyrstur has the masculine ending -ur, meaning a male is speaking. If a woman were saying this, she would use the feminine form and say Ég er þyrst.

I know the dictionary form of the verb "to want" is vilja. Why is it just vil in this sentence?

Vilja is a highly irregular verb. While many regular verbs in the "ég" (I) form end in a vowel (like ég tala or ég drekk), vilja loses its ending completely to become ég vil. It is an important exception to memorize!

How does vatnið mean "the water"? Where is the word for "the"?

Icelandic usually doesn't use a separate word for "the". Instead, it attaches an ending directly to the end of the noun. The base word for "water" is vatn (a neuter noun), and adding the definite suffix -ið turns it into vatnið ("the water").

What case is vatnið in, since it comes after the verb vil?

It is in the accusative case, because it is the direct object being wanted. However, because vatn is a neuter noun, its accusative form is exactly the same as its nominative (dictionary) form. That's why the word doesn't look like it changed case at all.