What kind of fish could live in a backyard pond?

Generally Koi.

You can purchase these in most large pet stores such as PETCO

If the pond is sheltered from birds of prey, you can keep most coldwater fish like koi and goldfish. You should set the pond up, get it running and stable and test the temperature in the coolest parts of the day and night to get a good idea of what fish you can buy with that tolerance. Most goldfish and kois and some of the larger catfish can be kept outside in late spring and summer. You’ll have to make arrangements to bring them in for fall, winter and early spring if you don’t live in a place that stays fairly steady year round.

Koi, goldfish, mosquito fish, minnows (white clouds are gorgeouse!), and tad poles whom become frogs!

Here’s a fantastic site for good and bad (poisonous) plants, fish and eggs for sale:

Best of Luck!

goldfish
some catfish
koi (check your laws)
Native fish (check your laws)
gambusia (check your laws)
White clouds (check your laws)