From a strictly-defined standpoint, alpha testing is acceptance testing, i.e. you try out a version of the software (or a part of it) and then report back your findings in terms of how you liked it. This is as opposed to beta testing, where the software you are testing is a bit more finalised and you're testing the robustness of it rather than the likability of the content and features themselves.
Well if we're going "strictly-defined" then one of the big differences between Alpha testing and Beta testing is that an Alpha test is supposed to be carried out at the developer's site. I did wonder when I saw it come up during the Kickstarter if those lucky souls were going to get a day out in Cambridgeshire.