Sanity testing or Build verification test(BVT):- Whenever 
we receive abuild from development team the basic features 
of the application is tested to verify the stability of the 
appication for further testing.

SMoke testing:- It is Similar like sanity test, this test 
is conducted to verify, are there any issues in Software, 
before releasing to the Test Team.

Sanity testing is done by Test engineer.

Smoke testing is done by Developer or White box engineers.

I think Prashant is right except for the fact that Sanity 
Testing is done before the product is to be launched to 
check whether after bug fixing the application is working 
fine.

when a new build is received from developers we do check 
for some basic functionality in the application(Ex : 
Navigating to all the tabs/links in the application).This 
is called smoke testing

It can be other way also when the final build is frozen and 
ready to be shipped into market some smoke testing is 
performed.

Sanity testing : Checking for the functionality of the 
module/application,whether it is working as expected is 
sanity testing.

Smoke testing is done by developers before the build is 
released or the testing done by testers before accepting 
the build.

Sanity testing provides whether it is reasonable to proceed 
with further testing