This is a term you may see in a sushi restaurant or other Japanese restaurant. It means "to entrust" or "I'll leave it to you." When entrusted with your meal, the chef usually will start from the lightest dishes to the most decadent. It doesn't necessarily mean just sashimi or sushi. It could be grilled or steamed items.
I love eating this way and, to be honest, cooking this way. I basically run our business this way. I don't have a huge file of menus for my clients to choose from. Instead, I write the menu just for them, and use the best ingredients of the season. I ask if there are any dietary restrictions or allergies and at what temperature you like your meat cooked.
If you get a chance to order Omakase style at a sushi restaurant, you trust you won't be disappointed and the chef will appreciate it, too. If you get lucky, you might find other restaurants that may be willing to do this for you if you are a frequent visitor. Or you could just have us come to your house!