For a shorter stay
For a shorter stay, hotels are the most convenient places to live in. You do not have to worry about groceries, cooking, cleaning, etc., and most of them have spas, gyms and a number of different services right where you are. They are comfortable, however, from a price point of view they might be a bit steep for longer periods.
Here is a link to a list of hotels in Budapest.
Quite similar to regular hotels, apartment hotels are built up of a selection of apartments, with the exception that they are fully equipped. People usually decide to stay at an apartment hotel if they are planning to stay for longer, or they want to take care of their own cooking, laundry and more.
Here is a list of apartment hotels in Budapest.
For a longer stay
AirBnB renting spread in Hungary around 2015 and has been present since. Most available houses are in the inner districts but you can find some flats further from the city centre too. If you are not familiar with AirBnB or do not like the concept, you might be better of looking for a place to rent for an extended stay.
For a longer stay we would recommend looking for a place to rent if you want to feel the full Hungarian experience.
With a quick search on the web, you can find various sites and companies that will help you find and choose the place you desire. You can filter your searches depending on the location, the type of flat, the number of rooms or the price you want.
Here are some examples that we found: