A slash is a swift, violent cut. As a result of being swift and violent, the wound created by a slash is longer and deeper than a shallow surface cut. To slash has therefore come to mean making deep cuts.
It is in this sense that we use it in the expression, slashing a budget. That is, rather than making small cuts, or gradual cuts, slashing a budget means making deep, dramatic cuts, ones that affect the near future greatly.
The noun form would be a budget slash (singular) or budget slashes (plural).
Example: On January 6, 2011, the Internet news portal The Drudge Report characterized this article on Pentagon spending cuts that would result in a reduction of U.S. active duty military troops as "Pentagon Budget Slash: Obama To Cut Troops On Active Duty". As is customary with this tabloid-like news site, the headline is intended to dramatize; after all, instead of "budget cut" we have "budget slash," implying a violent, damaging process.
Whether this characterization is accurate or not, this is how the writer wishes us to read and understand his headline. In that context, this is a good example of how "slashing" is used in a budgetary context.