When a president chooses his vice president he is not a president yet, he or she is still in a presidential campaign to try to get elected as president so towards the end of the campaign and when the...
Originally, the runner-up would be the Vice President. Now, however, the Republican nominee, and the Democrat nominee both choose a vice usually from the some party.
The President will nominate a new Vice President, who must be confirmed by both houses of the Congress in accordance with the 25th Amendment.(see related link)