Learn French: Il écrit un rapport sur cette affaire. - He writes a report about this matter.

Écrit comes from the irregular verb écrire (to write). In the present tense for il, elle, or on, it drops the -re ending and adds a -t to become écrit. The other singular forms end in -s: j'écris and tu écris.

I thought sur meant 'on'. Why is it translated as 'about' here?

You are right that sur usually means physically 'on' (like a book on a table). However, when talking about a topic or subject, such as a report, a movie, or a conversation, sur is used to mean 'about' or 'on the subject of'.

Since affaire starts with a vowel, shouldn't it be cet affaire?

Affaire is a feminine noun. The feminine demonstrative adjective (a pointing word meaning 'this' or 'that') is always cette, even when the noun starts with a vowel. You only use the special form cet for masculine nouns that start with a vowel sound.

Can affaire also mean 'business'?

Yes! Affaire is a very flexible word. Depending on the context, it can mean a matter, a case, a deal, or business. Here, 'matter' or 'case' makes the most sense because he is writing a report.