Here you can find pictures of columnar, crested, globular, grafted and leaf cacti. They should help you to identify them.
Cacti
Acanthocalycium glaucum
Origin: Argentina
Astrophytum myriostigma “Huboki”
Common name: Bishop’s Cap
Synonyme: Astrophytum columnare, Astrophytum myriostigma var. columnare
Origin: The wild type is native to Mexico
Astrophytum myriostigma “Nudum”
Common name: Bishop’s Cap
Origin: The wild type is native to Mexico.
Astrophytum ornatum var. glabrescens
Common name: Bishop’s Cap
Synonyme: Echinocactus ornatus
Origin: Mexico
Astrophytum ornatum var. mirbelii
Common name: Bishop’s Cap
Synonyme: Echinocactus ornatus
Origin: Mexico
Austrocylindropuntia subulata “Cristata”
Common name: Eve’s Needle, Eve’s Pin, Long Spine Cactus
Synonyme: Opuntia subulata “Cristata”
Origin: The wild type is native to Argentina, Bolivia and Peru.
Austrocylindropuntia vestita “Cristata”
Common name: Long Spine Cactus, Eve’s Pin
Synonyme: Opuntia vestita “Cristata”
Origin: The wild type is native to Argentina and Bolivia.
Cephalocereus senilis
Common name: Old Man Cactus, Old Man of Mexico, Bunny Cactus
Origin: Mexico
Cereus jamacaru “Spiralis”
Common name: Spiral Cereus, Queen of the Night
Synonyme: Cereus forbesii “Spiralis”, Cereus forbesii spiraliforme, Cereus peruvianus forma spirale
Origin: The wild type is native to Brazil
Cereus repandus “Florida”
Common name: Giant Club Cactus, Hedge Cactus
Synonyme: Cereus peruvianus
Origin: The wild type is native to Grenada, Netherlands Antilles, Venezuela.
Cereus repandus var. monstrosus
Common name: Giant Club Cactus, Hedge Cactus
Synonyme: Cereus peruvianus
Origin: The wild type is native to Grenada, Netherlands Antilles, Venezuela.
Cleistocactus strausii
Common name: Silver Torch, Wooly Torch
Origin: Argentina, Bolivia, Peru, Uruguay
Cleistocactus winteri “Cristata”
Common name: Golden Rat Tail, Rat Tail Cactus
Synonyme: Hildewintera aureispina “Cristata”
Origin: The wild type is native to Bolivia.
Corynopuntia grahami
Common name: Club Cholla
Synonyme: Grusonia grahamii, Corynopuntia planibulbispina
Origin: USA (New Mexico, Texas), Mexico
Coryphantha pycnacantha
Common name: Cory Cactus
Synonyme: Coryphantha andreae
Origin: Mexico
Cylindropuntia imbricata
Common name: Candelabrum Cactus, Cane Cholla, Walking Stick Cholla, Tree Cholla, Chainlink Cactus
Origin: USA to Mexico
Echinocactus grusonii
Common name: Golden Barrel Cactus, Golden Ball, Mother-in-Law’s Seat
Origin: Mexico
Echinocactus grusonii “kurzdornig”
Common name: Golden Barrel Cactus, Golden Ball, Mother-in-Law’s Seat
Origin: The wild type is native to Mexico.
Echinocactus platyacanthus
Common name: Giant Barrel Cactus
Origin: Mexico
Echinocereus coccineus
Common name: Scarlet Hedgehog Cactus, Arizona Hedgehog, California Hedgehog, Claret-Cup Cactus
Origin: USA to Mexico
Echinocereus reichenbachii ssp. baileyi
Common name: Hedgehog Cactus, Lace Cactus
Synonyme: Echinocereus baileyi
Origin: USA (Oklahoma, Texas)
Echinocereus viereckii ssp. morricalii
Common name: Hedgehog Cactus
Origin: Mexico
Echinopsis ancistrophora ssp. cardenasiana
Common name: Easter Lily Cactus
Synonyme: Echinopsis cardenasiana
Origin: Argentina, Bolivia
Echinopsis atacamensis
Common name: Pasacana Tree Cactus
Synonyme: Echinopsis pasacana, Trichocereus atacamensis var. pasacana
Origin: Argentina, Bolivia, Chile
Echinopsis candicans
Common name: Argentine Giant
Origin: Argentina
Echinopsis chamaecereus
Common name: Peanut Cactus
Synonyme: Chamaecereus silvestrii
Origin: Argentina
More infos: White, pink, orange and yellow flowering cultivars are available.
Echinopsis chamaecereus “Lutea”
Common name: Yellow Peanut Cactus, Banana Cactus
Synonyme: Chamaecereus silvestrii “Aurea”
Origin: The wild type is native to Argentina.
Echinopsis subdenudata
Common name: Easter Lily Cactus
Origin: Bolivia
Epiphyllum “Tantico”
Common name: Leaf Cactus, Orchid Cactus
Origin: Epiphyllum species are native to Central America.
More infos: Hybrid (crossing) between E. “Pistachio” x E. “Tassel”.
Escobaria sneedii
Common name: Sneed’s Pincushion Cactus, Carpet Foxtail Cactus
Synonyme: Coryphantha sneedii
Origin: New Mexico & Texas (USA)
Espostoa lanata
Common name: Peruvian Old Man Cactus, Cotton Ball Cactus, Snowball Cactus
Origin: Ecuador, Peru
Espostoa melanostele
Common name: Old Lady Cactus, Cotton Ball Cactus
Synonyme: Pseudoespostoa melanostele, Pseudoespostea melanostele
Origin: Peru
Ferocactus gracilis
Common name: Fire Barrel Cactus
Origin: Mexico
Ferocactus latispinus
Common name: Devil’s Tongue Barrel Cactus
Origin: Mexico
Ferocactus latispinus ssp. spiralis
Common name: Devil’s-Tongue Cactus
Synonyme: Echinocactus spiralis
Origin: Mexico
Ferocactus peninsulae
Synonyme: Ferocactus horridus
Origin: Mexico
Ferocactus pilosus
Synonyme: Ferocactus stainesii
Origin: Mexico
Frailea buenekeri
Origin: Brazil
Gymnocalycium anisitsii ssp. anisitsii
Synonyme: Gymnocalycium anisitsii ssp. damsii, Gymnocalycium damsii
Origin: Bolivia, Brazil, Paraguay
Gymnocalycium baldianum
Common name: Spider-Cactus, Dwarf Chin Cactus
Origin: Argentina
Gymnocalycium bruchii
Common name: Chin-Cactus, Spider Cactus
Synonyme: Gymnocalycium albispinum, Gymnocalycium lafaldense
Origin: Argentina
Gymnocalycium mihanovichii
Common name: Chin Cactus, Moon Cactus, Red Cap Cactus, Ruby Ball
Origin: Argentina, Paraguay
Hatiora salicornioides
Common name: Easter Cactus, Drunkard’s Dream, Dancing Bones
Synonyme: Rhipsalis salicornioides, Hatiora bambusoides
Origin: Brazil
Hatiora × graeseri
Common name: Easter Cactus, Whitsun Cactus, Holiday Cactus
Origin: The wild type is native to Brazil.
Hylocereus undatus
Common name: Pitahaya, Dragon Fruit
Origin: Mexico and cultivated in other tropical or subtropical regions.
Mammillaria decipiens
Common name: Bird’s Nest Pincushion
Origin: Mexico
Mammillaria elongata
Common name: Gold Lace Cactus, Ladyfinger Cactus
Origin: Mexico
Mammillaria elongata “Cristata”
Common name: Gold Lace Cactus, Ladyfinger Cactus
Origin: The wild type is native to Mexico.
Mammillaria hahniana
Common name: Old Lady Cactus, Old Lady Pincushion
Origin: Mexico (Guanajuato, Querétaro, Tamaulipas)
Mammillaria spinosissima “Un Pico”
Common name: Spiny Pincushion Cactus
Origin: “Un Pico” is a cultivar, Mammillaria spinosissima is native to Mexico (Guerrero, Morelos).
Specifics: Grows globular only as a young plant, columnar form shows up only with age.
Mammillaria vetula ssp. gracilis
Synonyme: Mammillaria gracilis
Origin: Mexico
Matucana madisoniorum
Common name: Melon Cactus
Origin: Peru
Myrtillocactus geometrizans
Common name: Bilberry Cactus, Whortleberry Cactus, Blue Candle, Blue Flame
Origin: Mexico
Opuntia elatior
Common name: Prickly Pear
Synonyme: Opuntia bergeriana
Origin: Colombia, Costa Rica, Lesser Antilles, Netherlands Antilles, Panama, Venezuela
Opuntia ficus-indica
Common name: Prickly Pear, Indian Fig Opuntia, Barbary Fig, Cactus Pear
Origin: Probably Mexico
Opuntia humifusa
Common name: Eastern Prickly Pear, Indian Fig
Origin: Canada to Mexico
Opuntia microdasys
Common name: Bunny Ears, Bunny Cactus, Angel’s-Wings, Polka-Dot Cactus
Origin: USA (Texas) to Mexico
Opuntia microdasys “Albata”
Common name: Bunny Ears, Bunny Cactus, Angel’s-Wings, Polka-Dot Cactus
Origin: The wild type is native from USA (Texas) to Mexico.
Opuntia microdasys “Rufida”
Common name: Bunny Ears, Bunny Cactus, Angel’s-Wings, Polka-Dot Cactus
Origin: The wild type is native from USA (Texas) to Mexico.
Oreocereus trollii
Common name: Old Man of the Andes Cactus
Origin: Argentina, Bolivia
Pachycereus marginatus
Common name: Mexican Fencepost Cactus, Organ-Pipe Cactus, Organ Cactus
Synonyme: Cereus marginatus
Origin: Mexico
Pachycereus pringlei
Common name: Mexican Giant Cardon, Elephant Cactus
Origin: Mexico
Parodia mammulosa ssp. submammulosa
Common name: Tom Thumb Cactus
Synonyme: Notocactus submammulosus
Origin: Argentina, Brazil, Uruguay
Pilosocereus pachycladus
Common name: Blue Torch Cactus, Blue Columnar Cactus
Synonyme: Pilosocerus azureus
Origin: Brazil
Polaskia chichipe
Common name: Chichibe, Chichipe, Chichitun, Chichituna
Origin: Mexico
Rebutia “Sunrise”
Common name: Crown Cactus
Origin: Rebutia “Sunrise” is a crossing between Rebutia heliosa and Rebutia pulvinosa ssp. albiflora.
Rebutia heliosa var. cajasensis
Synonyme: Aylostera heliosa var. cajasensis
Origin: Bolivia
Rebutia marsoneri
Common name: Crown Cactus
Origin: Rebutia marsoneri is native to the Argentine province of Jujuy.
Specifics: The Crown Cactus stays small, growing only up to five centimeters tall and six centimeters wide.
Rhipsalis cereuscula
Common name: Mistletoe Cactus
Origin: Brazil
Rhipsalis crispata
Common name: Mistletoe Cactus
Origin: Brazil
Rhipsalis mesembryanthemoides
Common name: Mistletoe Cactus
Origin: Brazil
Rhipsalis pilocarpa
Common name: Mistletoe Cactus
Origin: Brazil
Schlumbergera truncata cv.
Common name: Christmas Cactus, Holiday Cactus
Origin: The wild type is native to Brazil.
Schlumbergera truncata cv.
Common name: Christmas Cactus, Holiday Cactus
Origin: The wild type is native to Brazil.
Stenocactus crispatus
Common name: Brain Cactus
Synonyme: Echinofossulocactus crispatus, Echinofossulocactus kellerianus
Origin: Mexico
Sulcorebutia rauschii
Origin: Bolivia
Tephrocactus articulatus
Common name: Spruce Cone Cactus, Paper Spine Cholla Cactus
Origin: Argentina
Thelocactus bicolor
Common name: Straw Thelocactus, Glory of Texas
Origin: USA (Texas), Mexico (Chihuahua, Coahuila, Durango, Nuevo León, Tamaulipas)
Thelocactus rinconensis
Common name: Bird’s-Nest Cactus
Origin: Mexico (Coahuila, Nuevo León)
Thelocactus setispinus
Common name: Miniature Barrel Cactus, Hedgehog Cactus
Synonyme: Hamatocactus setispinus
Origin: USA (Texas), Mexico (Coahuila, Nuevo León, Tamaulipas)