The Federal Housing Administration (FHA) is to lenders what FDIC insurance is to savers; it protects lending institutions from mortgage defaults.
By protecting lenders, the FHA was begun with the intention to stimulate the housing market. The FHA was established in 1934 in an effort to stimulate the construction and purchase of new homes by offering insurance protection to the institutions (banks and mortgage companies) who make mortgage loans.
With the added protection, lending institutions are more likely to loan money to more clients. This has a downside, of course: by shifting risk to the FHA, the bank can avoid having to answer for some loans that it perhaps knew were not wise to approve.
Defaults on mortgage payments are a large part of what eventually caused the meltdown of 2008, since too many pools of mortgages were given high credit ratings and large financial institutions became overweight in them for their high yields.
The FHA is not really to blame, of course, and it has done much more good than harm.