Counter-part Pokémon are Pokémon that have some kind of link to each other but are usually unrelated in terms of evolution. Many of these counter-parts are legendaries or version exclusives.
Known Counter-part Pokémon
Generation I
Caterpie and Weedle
Both are bug types with a 3 stage evolution. They are very similar in looks and appear in the same games. Version exclusives in Black and White.
Ekans and Sandshrew
They both have a 2 stage evolution and version exclusives.
Nidoran♂ and Nidoran♀
While they are almost the same, they are classified as two seperate Pokémon and their evolutions are counter-parts being Male or female only.
Clefairy and Jigglypuff
Both have a gender ratio of 1 male to 3 females, are purely normal types, can both learn Sing, and evolve with the Moon Stone.
Vulpix and Growlithe
They evolve with the Fire Stone and are version exclusives.
Meowth and Mankey
They have a 2 stage evolution line and evolve at Level 23. Version exclusives.
Grimer and Koffing
They are Poison-types that represent pollution.
Shellder and Staryu
They have a 2 stage evolution line and evolve with the Water Stone.
Hitmonlee and Hitmonchan
Until it was revealed that they had a common pre-evolved form and another alternative evolution, they were considered counter-parts.
Electabuzz and Magmar
Version exclusives. When pre-evolved forms for Jynx, Magmar and Electabuzz were released, the three could have been considered a trio of counter-parts until Electivire and Magmortar were released, removing Jynx and Smoochum from the picture.
Omanyte and Kabuto
They are the first fossil Pokémon ever introduced.
Mewtwo and Mew
The two legendary Pokémon have similar attacks, abilities and appearances as well as the back-story of Mewtwo being a clone of Mew.
Generation II
Murkrow and Misdreavus
Both were version exclusive in Diamond and Pearl and Black and White. Both evolve with a Dusk Stone.
Ledyba and Spinarak
Both are verision exclusive (with Ledyba in Silver and SoulSilver and Spinarak in Gold and HeartGold), have a two-part evolution stage, and are dual types when evolved.
Tauros and Miltank
Both resemble cows and can only be one gender. They are one of the less commonly acknowledged counter-parts but often appear next to each other in Region Pokédexes.
Espeon and Umbreon
Generation II's eeveelutions that evolve depending on the time of the day.
Steelix and Scizor
Both evolve with a Metal Coat.
Teddiursa and Phanpy
Version exclusives found in Johto.
Lugia and Ho-oh
These legendary version mascots are counter-parts, each appear in only one out of HeartGold and SoulSilver (until you complete the main story) and both are leaders of Legendary Trios (Lugia=Articuno, Zapdos, Moltres and Ho-oh=Raikou, Entei and Suicune)
Generation III
Silcoon and Cascoon
Although evolved from Wurmple they are counter parts because one evolves into a Dustox (Bug/Poison) and the other into a Beautifly (Bug/Flying).
Lotad and Seedot
Both are Grass-types, version exclusives and evolve under the same conditions (to get to their first stage, they need to level up to level 14 and to their final stage, they need to have a type of stone used on them).
Sableye and Mawile
Version exclusives that have no evolutions.
Ninjask and Shedinja
They both evolve from Nincada and represent life & death.
Plusle and Minun
Despite having no relation in terms of evolution, they look almost identical and their abilities only work when used together. Version exclusives in Black and White.
Volbeat and Illumise
Both are Firefly Pokémon and can only be one gender.
Zangoose and Seviper
The first example of counter-parts that don't very many asthetic similarities or have some kind of friendship. In fact they hate each other and fight all the time as seen by their claw marks on each other. Both have similar markings and are version exclusive.
Solrock and Lunatone
They are both genderless, both the same type, both version exclusive and represent the sun and moon, which are opposites.
Huntail and Gorebyss
Both evolve from Clamperl and are traded with a Deep Sea held item.
Latias and Latios
Another example of version and gender exclusives.
Groudon and Kyogre
Version exclusive version mascots which represent the land and the sea. Another example of a red and blue Counter-part.
Generation IV
Gardevoir and Gallade
Although both evolve from Kirlia, they are both very gender orientated, Gardevoir being feminine and Gallade being masculine.
Cranidos and Shieldon
Fossil Pokémon that are version exclusive and have a two-part evolutionary stage.
Stunky and Glameow
Both version exclusive (Stunky in Diamond and Glameow in Pearl) and both are used by Team Galactic's Jupiter and Mars respectively. They are also both a two stage evolution. Both Pokémon and their evolutions are unavailable in Platinum.
Bonsly and Mime Jr.
Both are version exclusive and evolve in the same way.
Dialga and Palkia
Version mascots for Diamond and Pearl control Time and Space which are often used together.
Cresselia and Darkrai
Both are related to the moon, Darkrai causes bad dreams and Cresselia cures them.
Phione and Manaphy
They both represent the sea and they are parent and child that cannot evolve.
Generation V
Throh and Sawk
Both are Fighting-type and look similar as if wearing a judo uniform with a black belt.
Sewaddle and Venipede
Both Pokémon of a 3-stage evolution.
Cottonee and Petilil
Both are version exclusive Grass-types that evolve with the Sun Stone.
Gothita and Solosis
Both are Psychic-types in a 3 stage evolution. Version exclusives.
Karrablast and Shelmet
These Pokémon can only evolve when traded with each other.
Braviary and Mandibuzz
A male and female counter-part pair who are based on American settlers and native Americans. Version exclusives.
Heatmor and Durant
These are rivals like Seviper and Zangoose but are NOT version exclusive.
Reshiram and Zekrom
The ultimate counter-parts represent Yin and Yang. Version exclusives.