Here is the full list of all 40 Fish on the desktop achievements.
Arapaima
Atlantic cod
Barracuda
Black redeye fish
Black spotted grunt
Butterfly fish
Carp
Clownfish
Discus v1
Discus v2
Discus v3
Electric catfish
Emperor angelfish
Foureye butterfly fish
Frontosa
Gold redeye fish
Great white shark
Koi
Mackerel
Mahi mahi
Mandarin fish v1
Mandarin fish v2
Moorish idol
Nimbochromis
Orangespine unicron fish
Oreochromis
Peacock bass
Perch fish
Pike
Piranha
Powderblue tang
Rainbow cichlid
Red eye fish
Salmon
Siganus
Siganus guttatus
Siganus javus
Trewavas cichlid
Trout
Yellowtang