Draining is an effect that weapons with the draining brand, certain Necromancy spells, and certain monsters can inflict with their attacks. It withers its victim, reducing their general effectiveness, but its exact effect varies depending on the target.
When used against susceptible monsters it causes them to be permanently weakened, losing 1 HD, a few points of max HP, and making all of their attacks slightly weaker. Draining monsters repeatedly will stack these effects. While drained enemies are somewhat easier to defeat, they also give you less experience for defeating. However, this reduction is very minor and should mostly be ignored in your decisions.
All demons, undead, and plants are completely immune to draining, as are any natural monsters with at least one rank of negative energy resistance. One especially good target for draining attacks are shapeshifters. The forms these monsters can assume vary depending on their current HD. Draining their HD leaves them with a much less deadly set of options.
Draining attacks will inflict the Drained status effect on your character, temporarily reducing your effective skill levels across the board. Multiple hits will increase the severity of the draining; being lightly drained may be an annoyance as you lose 1 or 2 skill levels in all your skills, but severe draining can seriously reduce your attack speed, ability to cast spells, and general combat performance to dangerous levels.
This effect is not permanent, but will only disappear after you've gained sufficient experience. The worse the drain, the longer it'll take to work off the effect. You can reduce both the chance of being drained and the severity of draining by having negative energy resistance, with each rank reducing both by 33%.
Prior to 0.13, getting drained would damage your character's experience, potentially forcing you down a level. This was permanent, but you could simply gain more experience to make up for it. This had no impact on your skills.