You can use Enumerable#each_slice:
["cat", "dog", "mouse", "tiger"].each_slice(2).map(&:last)
# => ["dog", "tiger"]
Update:
As mentioned in the comment, last is not always suitable, so it could be replaced by first, and skipping first element:
["cat", "dog", "mouse", "tiger"].drop(1).each_slice(2).map(&:first)
Unfortunately, making it less elegant.
IMO, the most elegant is to use .select.with_index, which Nakilon suggested in his comment:
["cat", "dog", "mouse", "tiger"].select.with_index{|_,i| (i+1) % 2 == 0}