Learn Icelandic: Í dag er hann veikur.

This is because of the V2 rule (verb-second rule). In Icelandic, the main verb almost always has to be the second block of information in a sentence. Since this sentence starts with the time phrase Í dag (Today), the verb er (is) must step into the second slot. This pushes the subject hann (he) into the third spot.

Would it be wrong to say Í dag hann er veikur?

Yes, that would be grammatically incorrect. While English allows saying "Today he is sick" without changing the word order, Icelandic strictly requires the verb to stay in the second position. You must invert (swap) the subject and verb to say Í dag er hann veikur.

Can I change the word order to put hann first?

Absolutely! If you start the sentence with the subject, it becomes Hann er veikur í dag (He is sick today). Notice that the verb er is still in the second position, right after Hann.

Why does the word for sick (veikur) end in -ur here?

Adjectives in Icelandic change their endings to match the gender of the noun or pronoun they describe. Because veikur describes hann (he), it must be in the masculine form. For most masculine adjectives describing the subject of a sentence, the default dictionary form ending in -ur is used.