Basically, what you can do is go under the main menu, select the magic tab and click Active Effects. If you currently have a disease, the effects will show up here in red (connoting a negative effect). Here are the diseases that you can contract in Skyrim as per the Elder Scrolls Wiki:
Ataxia: Lockpicking & Pickpocket become 25% harder. This disease is contracted from traps and Skeevers.
Black Heart Blight (Dragonborn DLC only): Drains 10 points from carry weight.
Bone Break Fever: -25 Stamina. Contracted from Traps, Bears.
Brain Rot: -25 Magicka. Contracted from Traps, Hagraven
Droops (Dragonborn DLC only): One-handed and two-handed weapon damage is 15% lower. Contracted from Ash Hoppers.
Rattles: Stamina recovers 50% slower. Contracted from Traps and Chaurus
Rockjoint: 25% less effective with melee weapons. Contracted from Traps, Wolves and Foxes
Sanguinare Vampiris: -25% health, progresses to Vampirism. Contracted from Vampires
Witbane: Magicka recovers 50% slower. Contracted from Traps and Sabrecat.
It depends on where you got it from. The best way to tell which disease you have is to (if your on the 360 like i am) press B switch over to Magic, and go to Active Effects. It will tell you which kind of disease you have.