There are several Budapest apartment rental websites where you also can find there houses, properties, holiday houses or business premises. Also you need to distinguish between long term and short term rental which makes a great difference in where to start your search. Assuming that you are interested in long term rentals I can recommend you the following websites to start your research:
- realestatehungary.hu (English version of ingatlan.com)
Except for Ingatlanbazar and Duna House (dh.hu) all websites have a English version however the property description will still be in Hungarian but at least you can see the features like size, number of rooms, condition etc in English.
For short term rentals like a week or even a month I still recommend you to check out the apartments listed on Airbnb which are perfect and uncomplicated for short term rentals and fairly cheap as well. Also have a look at the 10 most beautiful Airbnb apartments in Budapest I have collected recently.