This is a partial list of Designated State Butterflies
I am still search the web for the rest.

Alabama >> Tiger Swallowtail
Papilio glaucus Linnaeus is a very familiar species all throughout the country. The species pictured above ranges from Canada, down to Florida, and into Mexico (there are different forms of the tiger swallowtail in the western states and up into Alaska, i.e., they are typically smaller in size and more pale in color - yellow)

Arizona >> Two-Tailed Swallowtail
(Papilio multicaudata) is a large North American butterfly. Its appearance is similar to Western and Eastern Tiger Swallowtails. The Two-Tailed Swallowtail has distinctive yellow wings with a black tiger striping. Each hindwing has several blue markings (top and bottom) and 2 tails.

Arkansas >> Diana Fritillary
(Speyeria diana) butterflies are large and beautiful. Males are dark brown and orange. The much larger females are black and blue.
 

California >> California Dogface
Zerene eurydice  The California Dogface is named for the pattern on the forewings of the male: they are yellow with a black border outlining a profile shaped like a dog's face. A black spot on the wing forms the "eye" of the dog. The dog-head may be tinged with violet or pink iridescence. The adult female is all yellow except for a black spot in the forewing. In both sexes, the underside is yellow or green with a black circle instead of a spot.

 

Colorado >> Colorado Hairstreak
(Hypaurotis crysalus) Upperside dark purple with wide dark border and orange spots at lower outer edge of each wing. The Colorado Hairstreak Butterfly adopted as the official state insect on April 17, 1996. The butterfly is two inches in width and has purple wings with black borders, orange accents in the corners and blue on the underside. It may be found on both sides of the Continental Divide at elevations of 6,500 to 7,500 feet, in its usual habitat of scrub oak ecosystems.

Florida >> Zebra Longwing
Heliconius charitonius) are found in Florida.  Zebras fly slowly and don't startle easily, making them easy to follow and observe. A zebra resting at dusk can be gently coaxed to climb on your finger and to return, unflustered, to its perch. Zebras roost in groups, returning to the same location each night.
 

Georgia >> Tiger Swallowtail

Idaho>> Monarch
(Danaus plexippus) is a well-known North American butterfly.  Related species in the family are found on all continents except the polar regions, wherever milkweed and related plants are found. It also provides the Monarch with an intriguing form of protection, since the milkweed juices assimilated by the Monarch make it poisonous to predatory birds.

Illinois >> Monarch

Kentucky >> Viceroy
(Limenitis archippus) He's evolved to look like the poisonous monarch to the right so that predators will avoid him too!  Still can't tell them apart?  Notice that black stripe on the bottom wings of the Viceroy.  That's the give-away!  It's easily seen in the photos, but birds flying overhead looking for lunch cannot distinguish between the two.

Maryland >> Baltimore Checkerspot Butterfly
(Euphydryas phaeton) is a medium sized butterfly with dazzling coloration. Named after the early American colonist George Calvert, the first Lord Baltimore (whose crest was orange and black), the Baltimore Checkerspot is mainly black with orange and white spots. Its underwings are also orange, black, and white; with these three colors arranges in a checkered-like pattern.
 

Minnesota >> Monarch

Mississippi >> Spicebush Swallowtail
Eastern states from southern Canada to Florida; west to Oklahoma and central Texas. Occasionally strays to North Dakota, central Colorado, and Cuba

Missouri >> Spicebush Swallowtail

Montana >> Mourning Cloak
(Nymphalis antiopa), a velvety black butterfly with a two-inch wingspan. A bright yellow band lines the entire edge of the butterfly’s jagged wings, just outside of a row of iridescent blue spots.

New Hampshire >> Karner Blue
(Lycaeides melissa samuelis)  small butterflies about the size of a nickel. Males have a vibrant, silvery blue color on the upper surface of their wings. The upper surfaces of the females' wings are blue close to the body, fading to grayish-brown towards the edges.

New Jersey >> Tiger Swallowtail

Ohio >> Tiger Swallowtail

Oklahoma >> Black Swallowtail
Papilio polyxenes, is a butterfly found throughout much of North America. It is the state butterfly of Oklahoma.

Oregon >> Old World Swallowtail
Papilio machaon has a reddish-orange and blue eyespot on the upper side of the hind wing near tail with black along the lower border on the lower edge of hind wing not overlapping the eyespot. It can been found in open hilltops, mountain meadows, subarctic, boreal, northern prairie habitats, tundra and badlands in North America, South Alaska, British Colombia and Quebec and is the only swallowtail with circumpolar range

South Carolina >> Eastern Tiger Swallowtail
Papilio glaucus, is a large (12 cm wingspan) swallowtail butterfly. It is found in the Eastern United States, as far north as southern Vermont, and as far West as extreme Eastern Colorado. It flies from spring through fall, and most of the year in the southern portions of its range, where it may produce two or three broods a year.

Tennessee >> Zebra Swallowtail
Papilio marcellus Cramer is the most abundant regular North American representative of the kite swallowtails, named for their triangular wings and long sharp tails. Despite a large range, the Zebra occurs only near pawpaw or its relatives
 

Texas >> Monarch

Vermont >>  Monarch

Virginia >> Tiger Swallowtail

West Virginia >> Monarch