What produces the strong odor of decaying dead animals?
Two compounds are specifically associated with the smell of decaying cadavers (Hoffman et al., 2009), namely: 


1,5-pentanediamine (cadaverine), and 
1,4-butanediamine (putrescine)


Cadaverine is produced by breakdown of lysin and putrescine by the breakdown of ornithine (Science ME). Both are nitrogen-rich compounds and particularly pungent-smelling.


Source: Science ME

Both compounds are also responsible, in part, for the smell of bodily fluids such as urine and semen, smell of rancid butter, bad breath and stinky cheeses (Science ME).

Forensic Science Central UK mentions that after death, the body begins to decay after about 5-11 days (after an initial phase of bloating). The putrid smells subside mostly after 10-24 days, although a rancid cheesy smell might persist due to butyric acid. Note that many factors will affect decomposition rates and will affect this time course. Moreover, trained police dogs may be able to sniff out minute amounts of putrescine and cadaverine long after humans cannot detect it anymore. 

Note that, given the complex composition of animals and the complexity of the decomposition process, a host of volatile organic components are released during decomposition besides putrescine and cadaverine.

Reference
- Hoffman et al. Forensic Science International 2009; 186: 6-13