Creatures, spells, and traps can infect you with diseases. Diseases may have a range of debilitating effects that range from light to severe. Open the magic menu and check the Active Effects list to see which diseases you are infected with and what their effects are.
To cure yourself, drink a potion of Cure Disease or seek healing at a temple or shrine. The effects of diseases are cumulative, so they will add up over time if left untreated.
The following is a list of Diseases that may be contracted in Skyrim:
Ataxia: 25% penalty to Lockpicking, Pickpocket
Bone Break Fever: 25 less Stamina
Brain Rot: 25 less Magicka
Rattles: 50% slower stamina regeneration
Rockjoint: Melee weapons 25% less effective
Sanguinare Vampiris: 25 less health, progresses to Vampirism
Witbane: 50% slower Magicka regeneration
Watch the screen carefully whenever you are in combat to spot warnings that you have contracted a disease, especially when fighting rats, skeevers, bears, foxes and wolves, as well as when stepping on any traps.
When your character contracts a diseases, drink a cure disease potion if you have one, or go to a shrine and activate. Note that leveling up does NOT cure any diseases. It is best to have a few Cure Disease potions on you, as more often then not you will be contracting the disease in the middle of a mission and will not have a chance to attend a shrine for a while.
It is VERY easy to miss the notice saying that you have contracted a disease in the heat of combat. Once in a while, you should bring up the Magic menu and look under "Active Effects" to make sure you do not have a disease. Another indicator that you have caught a disease is if one of your skills is in Red. For example if your pickpocket skill or stamina is in red in the skills panel, it means you could have Ataxia or Bone Break Fever. One last indicator to look for, is that when you talk to NPCs they will sometimes tell you that you look sick or ill. This is another indicator that you have contracted one of the many diseases. Noticing and taking care of the disease as soon as possible is important because depending on your class and the diseases, your character may be severely gimped until the disease is cured.
The following races get a 50% resistance bonus to diseases: Argonian, Wood Elf (Bosmer)