Learn Icelandic: Ég er að hugsa um stelpuna. - I am thinking about the girl.

To show that an action is happening right now, Icelandic uses a structure very similar to English: the verb vera (to be) + að + the infinitive of the main verb. Here, ég er means 'I am', and að hugsa means 'to think', giving us 'I am thinking'.

Why doesn't the verb hugsa change its ending to match ég?

Because it is part of the vera að continuous structure! In this pattern, only the helping verb vera gets conjugated (becoming er to match ég). The main action verb hugsa (to think) simply stays in its infinitive or dictionary form.

Why is the word for 'the girl' stelpuna instead of stelpan?

The preposition um (about) forces the noun that follows it into the accusative case. The basic subject form for 'the girl' is stelpan, but when it changes to the accusative case because of um, the ending shifts, making it stelpuna.

Does the preposition um always mean 'about'?

Very often, yes! When you are talking or thinking 'about' a topic or a person, um is exactly what you need. It can also mean 'around' (like walking around a town), but in all of these uses, remember that um always triggers the accusative case.