Sanity Testing:-Tester conducts the Sanity test to ensure 
the stability of the application build.Tester finds weather 
the application build is stable for complete application or 
not.

Smoke Testing:-It covers the major functionality of the 
application without bothering with finer details.

Sanisetty Testing :
Testing of the Application or System on different Versions.
or 
The Combination of smoke and monkey testing.

Smoke Testing:
It covers the major funcionality of the Application.

SMOKE TEST :
                         Smoke testing is considerd as 
built level test.Each time the test team receives a new 
build(new version of program),an initial test is performed 
to determine that the most crucial functions of the program 
works and the build is stable enough and can be considered 
for further testing.

SANITY TEST :
                          Sanity testing is considered as 
release level testing.When software is ready to be shiped 
to client and we want to finally cheek if everything is 
fine and working as per requirement.It's a sub set of 
Regressing testing -a group of test cases are executed to  
cheek it is indeed working according to specification
(requirement).

  I hope this will do.But somebody may not agry with me as 
this defination will vary company to company... but i think 
this is the basic defination...

Sanity Testing; Test the application making sure that
critical and blocker Bugs are not reproduced in the latest
Build.

Smoke Testing; Test the application Functionality

Both Sanity testing and Smoke testing will be done for 
testing the functionality of the program whether the 
particular feature or module is functioning according to 
its specification or not. But Smoke testing  is superior to 
Sanity testing ., because When we take a web based 
application Smoke testing is done to check whether the URL 
is doing its functionality or not, where as Sanity testing 
will check individual module in it are functioning or not.

Smoke testing look only for major functionalities but 
Sanity testing will look in dept of each module/feature.