According to what we understand EA sent out this select Email to a select number of users, but we are not sure how they were chosen. If you were invited already, you are not going to get invited. While EA will not comment on the number of invites that were sent out or how long this Alpha test is expected.
When the Email first arrived, many ignored it thinking that it was simply a scam, but it turns out it is actually legit. So if you were invited, you should respond quickly if you still can.
As for the possibility of a Beta test, that remains a mystery and only Respawn knows if they are planning one or not. If they are planning one, they still have no comment on it at this time as you can see above.