Its basically a variation on Worcestershire sauce. But thicker and fruitier. Similar to Jamaican pick-a-pepper and Japanese tonkastu sauce. The closest American equivalent would be steak sauce. A-1 or Heinz 57.
But brown sauce is much thicker and a bit sweeter. Americans tend to know HP and Chef, the major Irish brand if they have any experience with the stuff.
It's waaaaay better than ketchup though. Great on a burger and acceptable on dogs in my opinion. Mostly because it's not basically red sugar.