A higher Alchemy skill improves the effectiveness of the potions and poisons you create using the skill.

Every character can benefit from Alchemy, because even if you do not invest in the perks you can still use the skill to generate all sorts of meaningful potions.  Should you use weapons, you can also create poisons from the inconveniencing to the deadly.  Practically all characters will want the perks, though, because they enable you to create much more powerful and thus much more useful potions and poisons to keep yourself alive and make your enemies dead.

Level up your Alchemy skill by discovering the effects of ingredients (either by mixing them to make new concoctions or eating them to see what they do to you) and by crafting potions and poisons (even if you already know what you’re going to get).  The fastest way, of course, is to use ingredients to create new mixtures, as you gain experience both from crafting and from discovery at the same time.  As with all other skills, trainers exist and you can gain up to five additional ranks each level by paying money for training.

TRAINERS
Journeyman: Lami (Morthal)
Expert: Arcadia (Whiterun)
Master: Babette (Dark Brotherhood)

PERKS

Alchemist (1/2/3/4/5)
Your concoctions are 20%/40%/60%/80%/100% stronger.
Requires Alchemy 0/20/40/60/80

Physician
Your potions that restore Health, Magicka, and/or Stamina restore 25% more of these resources.
Requires Alchemy 20, Alchemist 1

Benefactor
Your potions’ positive effects are 25% stronger.
Requires Alchemy 30, Physician

Poisoner
Your poisons’ negative effects are 25% stronger.
Requires Alchemy 30, Physician

Experimenter 1/2/3
Eating an ingredient reveals the first two/the first three/all four of its effects.
Requires Alchemy 50/70/90, Benefactor

Concentrated Poison
When you apply a poison to your weapon, it lasts for twice as many hits (usually two hits instead of one).
Requires Alchemy 60, Poisoner

Green Thumb
When you gather ingredients from a plant, you get two ingredients instead of one.
Requires Alchemy 70, Concentrated Poison

Snakeblood
You gain 50% poison resistance.
Requires Alchemy 80, Experimenter 1 OR Concentrated Poison

Purity
Your potions lose any negative effects and your poisons lose any positive effects.
Requires Alchemy 100, Snakeblood