That's a difficult question to answer! In terms of sentimental value, it would probably be our sideboard which is a family heirloom - my mum inherited it from her parents, they were given it by my grandpa's parents.... and my parents gave it to us as a wedding present. It's also our most valuable piece of furniture.
In terms of functionality and comfort though, it would be a toss up between my glider chair (about the only chair I can sit in comfortably, given my spine problems) and my bed.
The most important piece of furniture is definitely my bed. I could manage without other furniture or make do eg bricks and boards for book shelf, an old door or piece of wood resting on bricks for a table or an old card table from somewhere but I need a proper bed with inner spring mattress. Lots of other furniture is available second hand and can be done up or else cheaply as flat pack.