やって in "The film was on TV yesterday"
そのえいがは きのう テレビで やって いました。
I keep getting this sentence wrong because I don't understand what やって means or what its role in the sentence is.
Could someone explain?
Basically it’s the same やっている that means (doing), but it can also be used to indicate run, operate (in both transitive and intransitive form), work as/do the work of, or other “shades” of taking action, much like (do) in English.
店をやっています: Running a shop.
店がやっています: Shop is “operating”, or being open for business (short or long term, can be used for both).
教師をやっています (or sometimes 教師やっています): Work as a teacher.
So, the sentence you mentioned meant (the movie was running on TV yesterday).
Actually, it is 遣る, but it has several meanings, and (dispatch/send) are among them. Check out entry no. 6 in the link below, it corresponds to (do):
The sentence with only いました would be incomplete, because いました here is only an addition used to change the verb from its simple present tense (やる = run) to the past continuous tense （やっていました = was running).
That is true, and it's also a colloquial way of saying あげる (give). I read that one of the common mistakes made in Japanese business language is to use やられる as the respectful form of やる (a verb is changed into its respectful form following the same rule to change it into the passive form). This word usage is considered a mistake because since やる is originally a colloquial word, even its respectful form is not appropriate to use in polite language.