I used to think "Bundt cake" referred to a specific dessert, but it literally means any cake shaped in a ring-shaped pan. That said, it does tend to mean just one traditional recipe to a given person or family, so when someone says this, he usually means "that thing my aunt always makes for company."
This style of cake became popular in the US in the 50s and 60s, which is when home cooks got really excited about shortcuts. The advantage of baking a cake in this pan is that the pan does the decorative work for you. A Bundt cake is pretty on its own, prettier with powdered sugar or a glaze on top, and doesn't require the splitting, trimming, or other fussy steps of assembling a layer cake. It can also be served warm, which much shortens the time from oven to table.
The downside is that, unlike with layer cake, breaks are very hard to disguise. If a Bundt cake doesn't comes out of the pan in one smooth piece, you're screwed on presentation (unless you turn it into trifle, which is never a bad idea). For this reason, most Bundt cake recipes emphasize extreme greasing and flouring of the pan.
Because these cakes aren't meant to be paired with a lot of frosting, they tend to be richer than most other cakes. Recipes which specify a Bundt pan are often essentially pound cake, or include a baked-in filling such as a "tunnel of fudge," and I'm sorry I had to type that.