Hungary’s unit of currency is the forint.
Despite being a part of the European Union, Hungary does not use the euro. Ninety percent of the places you do business with in Budapest don’t accept euros. Outside the capital city, I wouldn’t plan on being able to use euros at all.
Some of the places that cater to tourists will take euros, and you might be able to use them to, say, put a down payment on an apartment, but for the most part I’d convert your euros, pounds or dollars to forint* (the national currency of Hungary) if you’re going to be in Hungary for more than a few hours.
*The name “forint” is derived from the city of Florence, Italy. Charles I of Hungary began using the florentinus in the 14th Century, and the name eventually morphed into forint. I learned this from listening to the super helpful Let’s Learn Hungarian podcast.
Even at the Budapest Airport, where most places accept euros as payment, you’ll get change in forint.
In Budapest, change windows are all over the place, and they all prominently display their exchange rates, so you can decide whether you’re getting a good deal. Just be sure to check the current exchange rate (and be aware that almost all change booths will charge a commission of 3 or 4 forint per dollar).
Most of the cash machines in town will let you withdraw money in both euros and forint.
By the way, it’s worth noting that most places don’t accept credit or bank cards. Grocery stores take cards, as do BKK ticket machines, but most other places you spend money will want to see cash. And they’ll want you to pay with the smallest bills or coins possible. At many places, such as bakeries, discount stores and flower shops, you’ll get a serious evil eye (and your money might even be refused) if you try to pay for items totaling 1190 HUF with a 10000 HUF bill. Most Hungarians carry coin purses for situations like this.