Alpha Testing: This is one type of UAT(User Acceptance Testing) which is done by client at developers site for mainly software applications.Here the application development is complete but still there might be any change done.
Beta Testing:This is one type of UAT(User Acceptance Testing) which is done by End-users at Clients/Customer's site for mainly software Products.Here the development of the product is completed(mandatory) but still some defects are need to be fixed.
Alpha testing is simulated or actual operational testing by potential users/customers or an independent test team at the developers' site. Alpha testing is often employed for off-the-shelf software as a form of internal acceptance testing, before the software goes to beta testing.  Beta testing
Beta testing comes after alpha testing and can be considered a form of external user acceptance testing. Versions of the software, known as beta versions, are released to a limited audience outside of the programming team. The software is released to groups of people so that further testing can ensure the product has few faults or bugs. Sometimes, beta versions are made available to the open public to increase the feedback field to a maximal number of future users.