Let's assume I'm working in Austria and I get fired. If I move to another european country, will I continue to perceive my unemployment money?
It is the Regulation (EC) No 883/2004 that is relevant here.
In principle the workland principle applies. These means that social security from the country where you work is relevant. When you are unemployed it is the social security of the homeland that applies. As @gael already mentioned there is an allowed period where you can live abroad to seek work:
An unemployed person may move to another Member State in order to seek work while retaining entitlement to benefits for three months. The competent services or institutions may extend this period up to a maximum of six months. If the unemployed person does not return on or before the expiry of this period he/she loses all entitlement to benefits.
The distinction between workland and home land might seem to be nonsensical at first sight, but it makes sense if you consider that in the EU many cross borders on a daily base to work. I my self live in one country and work in the other. For all the social security, while I am working I need to consult the social security in my working country. If however I become unemployed, I need to consult the social security of the country where I reside.
Generally speaking, you are getting your unemployment benefits from your country of residence. To be eligible, however, you must usually stay in that country (they send you job offers, check on you etc.). This is not transferable to another EU country, i.e., you can't just move to a different country and start getting the support there. Nevertheless, as Gaël Laurans points out in his comment it is possible to have that money sent also to another country. Just keep in mind to follow the procedure in the link - it is still important that you can be reached by the corresponding national employment authorities.