You could just as one question, "what Pokemon work well in what kind of weather".
In any case:
Vaporeon will be able to fully restore its HP by using Rest, and it will wake up at the end of its turn. This means that it doesn't need one move to replenish HP by using Wish, and it will also be cured of any status problems (bar confusion).
Ability: Dry Skin
This will heal it more than Black Sludge/ Leftovers will, giving it 1/8 of its HP back at the end of each turn. It is also immune to water moves.
Ability: Volt Absorb
Thunder gains 100% accuracy in rain, along with STAB. It will also be able to take electric hits itself, since its ability heals it when hit by an electric move.
Moves: Thunder, Hurricane
Dragonite gets the two moves that gain 100% accuracy in rain, and one with STAB. While its special attack lower compared to its attack, it's still at a base of 100.
Ability: Serene Grace
Moves: Thunder, Iron Head
In rain, the power of fire moves will be halved, so Jirachi will be able to take a hit from a fire STAB. Along with 60% of paralysing its opponent with Thunder, Jirachi will be able to make full abuse of Serene Grace, using Iron Head to paraflinch the opposing Pokemon.
Ability: Sniper/ Swift Swim
On a team in OU, Kingdra is a Pokemon with a very useful typing, being weak to only Dragon. Dragon Dance means it can abuse STAB Water moves like Waterfall, along with the added boost the move does in the rain.
While Kingdra's Swift Swim is banned on a team that has Drizzle in OU, it can have Swift Swim on a Drizzle team with Kyogre in Ubers, doubling its speed without the need of a Choice Scarf, meaning it can use a Life Orb or old a Lum Berry in order to set up Dragon Dance.
Ability: Swift Swim
Again, ability is only allowed in Ubers with Drizzle. Kabutops is a really frail Pokemon, but with a speed that barely works at a base level of 80, and an attack at a base of 115, Kabutops becomes a power sweeper. It also doubles as your rapid spinner, but then again that move is a bit useless on a Pokemon as frail as Kabutops.
Ability: Rain Dish
Typing: Water/ Poison
A popular OU Rapid Spinner, Tentacruel will be able to soak up Toxic Spikes while healing itself double the amount it usually can, holding Black Sluge and having Rain Dish. Poison typing means it's not weak to Grass, and base speed of 100 along with a base special defence of 120 means it's able to take a hit.
These are the main ones in OU (and two in Uber).
Generally, the Pokemon that have abilities which benefit them in rain and the two moves Thunder and Hurricane will be most commonly seen on a Drizzle team. I discounted Politoed from this, because it's a given that it will be seen on a rain team in OU. Weather teams that need Rain Dance to set up its offense tend to be a bit slow in getting anything done.