Learn Finnish: Lukiessani kirjaa metrossa tarvitsen istumapaikan. - When reading a book on the subway, I need a seat.

This is the active temporal construction, which replaces a kun (when) clause. It is built from the verb lukea (to read). You form the 2nd infinitive inessive (lukiessa-) and add the possessive suffix -ni to show that 'I' am the one reading. It literally functions like saying 'in my reading'.

Why is there no word for 'I' (like minun) before lukiessani?

Because the person reading is the same as the person who needs a seat. When the subject of the temporal construction is the same as the subject of the main verb (tarvitsen), you drop the genitive pronoun (minun) and only use the possessive suffix (-ni). If someone else were reading, you would need their genitive pronoun in front of it.

Both kirjaa and istumapaikan are objects in this sentence. Why is one in the partitive and the other in the genitive-accusative?

Kirjaa is partitive because the action of reading is ongoing and incomplete in this context; you are in the process of reading it. Istumapaikan takes the genitive-accusative (ending in -n) because it is a total object: you need one whole seat, and the verb tarvita (to need) takes a total object when referring to a specific, complete thing.

What are the parts that make up the word istumapaikan?