The citizens of Taiwan vote for their president every four years.

The students secretly voted their new teacher the best teacher in the school.

Why does the first in the above contain "for" while the second doesn't? What are the differences between "vote for" and 'vote?" Thanks.

The verb 'voted' can be used in different ways. Here are some comments.

The citizens of Taiwan vote for their president every four years.

In this context, the meaning is 'they elect their president every four years. Note that the name of the position is used, and not the name of a candidate.

Consider also this context.

They voted for Tom. The meaning here is that they gave their support to a candidate named Tom.

Note that the name of a candidate is used in this example, and not the name of the position.

It's also possible to use both,

eg They voted for Tom for President.

The students secretly voted their new teacher the best teacher in the school.

This use of 'voted' with a direct object means 'they elected or chose'. It is normally used only when speaking of the result of an election.

Do you need any more information about this?

Clive