Your cabin steward will make one every night - or almost every night- out of some of the clean towels he/she leaves you. Sometimes he will use a personal item, like sunglasses. You can leave them as is, if you want, and request additional towels, or take them apart. (I found if you leave one, you usually don't get a new one the next day.)
You can also buy small versions of them - really just stuffed animals with white terrycloth exteriors - in the gift shops.
Towel animals do not occur on all cruise lines; Carnival, Royal Caribbean and Disney provide them, but Holland America doesn't, and I never saw any on my NCL cruise either.
I refuse to ever go on a cruise line that doesn't offer towel animals!!
Seriously, I think it's one of the many fun things that makes cruising a blast. However, it's not unique to cruising. When we stayed at one of the Disney hotels, towel animals were made every day.
One last towel animal story: when we were on the Paradise last year, I took part in the "Survivor" competition. One of the challenges was to make towel animals. I almost won with my rendition of a duck-billed platypus (actually, it was pretty sorry-looking).
Just to let you know, last May, we went on Explorer of Seas, RC, and they had a class on how to make towel animals, along w/a brochure on how to make many of the different ones. We all loved it, and my kids are 19 and 21, so now we can make them at home if we want too! Great memories from our cruise!