Some of these are available as patterns, some are knit, some are crocheted, some are felted.
Where better to start than the heart? This one was created by Kristin Ledgett for Valentine's Day. Pattern is at http://www.knitty.com/ISSUEwinter08/PATTheart.php
Slightly more multimedia, this one even lights up and was made by Tatyana Yanishevsk
[added Aug 2013] this fluffy heart from Sarah Illenberger who makes objects for advertising use
And so to the respiratory system, these lungs were made by Anna Isherwood
Other organs include Gallbladder (this one by AnOptimisticCynic with removable stones...)
.. and Womb. The pattern from Knitty is here. This is the first thing I ever made in my own hand dyed and spun wool.
[Added Aug 2013 suggested by Poppy Cottage in the comments] There are a few variants of the booby baby beany for breastfeeding, this one is bySara Hanson
Holding it all up is the spine, this needle felted example from estonia76
[added 12/7/13] Suggested by Sue in the comments, the whole skellington
And helping it all move, the muscles. More pictures of this lovely work here
In charge of all of that is the brain, one here that you can wear on the outside (pattern by Alana Noritake)
.. and an anatomically correct one by Dr Karen Norberg, read about it here
[update Aug 2013] More brain and neurons here from CraftyHedhehog, a project called Neuroscience
[added Aug 2013] Another from Sarah Illenberger
On a microscopic level, some of us may actually have woolly chromasomes
[Added 12/7/13] Not for the squeamish, this is the entire digestive system, part of a knit-along some years ago, follow the links for the pattern:
[added Aug 2013] this digestive system from Sarah Illenberger
Another digestive system along with brain & nervous system and other organs, this project is called Body Box by Caroline Gates:
More visible bits and pieces include ears.
.. and eyes
This one's a cat toy!
Do you know of any more? Let me know in the comments.