The thing about burgundy is how differently it reads depending on the light in the room. Under warm bulbs it goes almost chocolatey. Under office strip lighting or outside on a grey afternoon, it can swing properly purple, which is not what you thought you were buying. I've got a dress like that and I've learned to only trust it in daylight before I decide how I feel about it.
Keep it away from red, obviously, but less obviously away from anything orange toned or pink toned too, because burgundy has a blue undertone hiding in it and those colours fight it rather than sit with it. Black is safe but can look a bit funereal, so I'd reach for camel, soft grey or a warm cream instead, they let the burgundy do something rather than just absorb it. As for when to wear it, this is an autumn and winter colour through and through, it looks slightly wrong in bright June sun but brilliant from October onwards, good for anything from a work do to a Sunday lunch to a wedding where you don't want to turn up in black. It flatters most skin tones too, which is more than you can say for a lot of jewel shades.
Fabric matters more with this colour than almost any other, because a cheap, thin weave makes burgundy look flat and a bit plasticky, like a school uniform jumper. A good one is usually a decent weight crepe, a brushed viscose, or something with a touch of wool in it for the colder months, all of which hold the colour with a bit of depth rather than letting it sit on the surface. Check the seams at the waist and under the arms, cheap burgundy dresses tend to pucker there because the dye process can make certain fabrics behave oddly under tension. Buttons and zips in a matching or darker thread are a good sign someone thought about finishing rather than just getting it out the door.
It ages well if you treat it properly, which mostly means not boiling it in a hot wash, burgundy dye can fade to a dusty pink faster than you'd expect and once that happens there's no getting it back. Cold wash, inside out, away from anything white, and it'll hold its colour for years rather than months. Mine's three winters in now and still looks like the one I bought, not a washed out cousin of it.
It's the one colour on the rail that makes black look like you didn't try.