Learn Turkish: Yeni sekreter listeyi kopyalasın. - Let the new secretary copy the list.

This is the 3rd person imperative. By adding the suffix -sın (or -sin, -sun, -sün depending on vowel harmony) to a verb stem, you are saying 'let him/her/it do it' or 'he/she/it should do it'. Here, we add it to the verb stem kopyala (to copy) to get kopyalasın.

Why is there a y and an i at the end of listeyi?

That is the accusative case, which marks a specific direct object. The word for list is liste. Because we are talking about the specific list being copied, it needs the accusative ending. Since liste ends in a vowel, we need the buffer letter y before adding the vowel harmony suffix -i.

Why does the suffix on kopyalasın use an ı (undotted i) instead of an i?

It follows 4-way vowel harmony. The last vowel in the verb stem kopyala is a. In Turkish, if the last vowel is a or ı, the suffix will use an ı. Therefore, the imperative suffix becomes -sın.

Could I translate this as 'The new secretary should copy the list'?