Both are belongs to UAT (User Acceptance Testing).
Alpha testing will be perfomed by the customer in developers
environment before delivering the product to the customer.
Beta testing will be performed by the users in
production/customer site after the delivering of project to
customer.

Alpha testing: Customer to Customer? I think it's not the
exact answer..

It's not "customer to customer".Alpha testing will performed
by the client/customer before accepting the product/project.

Alpha Test is done among the team members itself, it is the 
1st phase in testing.

Beta Testing is the testing where the audience tries the 
product, also called as Pre release testing

Alfa & Beta testing is basically connected with acceptance 
testing, but the envioroment is diffrent like in beta 
testing we test the software at client end, but in alfa 
testing client is invlove with developer envioroment.

Alpha and Beta testing is a part of the User Acceptance 
Testing.
But Alpha Testing is done when the application is nearly 
complete and so minor design changes can still be made here.
Developers,test engineers and the customer site 
representatives involve in this.
WHERE AS
Beta testing is testing an application when the development 
and testing are essentially complete and final problems and 
defects are to be sorted out.End-User typically involve 
here.
It is done at the client site.

Both alpha and beta testing are part of UAT(User Acceptance 
testing).
Alpha Testing : It is the test done by the client at the 
developer’s site. 
Beta Testing : This is the test done by the end-users at 
the client’s site.

Beta testing is a part of functional testing, which is done 
at  the client premises.

Alfa Testing is part of funtional Testing, which is done on 
lab.

Alpha Testing : Testing done by end user / customer within

the company in the presence of Developer before release.

Beta Testing : Testing done by end user at customers site 
in the absence of Developer Team , the application is 
uncontrolled but still not live.