Omakase is a Japanese word that literally means "entrusting." At least for gaijin, the word "omakase" is most commonly encountered in Japanese restaurants, where regulars will simply say "omakase" when asked for their order, and "omakase" is occasionally found as a choice on the menu. In this context, the word means "putting your trust in the chef," eating whatever the chef feels like cooking for you, and usually, paying whatever price the chef feels like charging.
While this might sound like a scary proposition, especially given some of the truly bizarre dishes that the Japanese have been known to call "food," an order of "omakase" arguably puts more pressure on the chef to create a worthy meal out of the freshest foods he has, to reward and retain a valued customer’s trust in him. As a general rule, customers who order omakase are possessed of among the most discerning and discriminating of palates, as the mere knowledge of the meaning of "omakase" partially indicates, and thus only the most skilled of chefs would dare to put omakase on the menu.
Moreover, despite the potential to be ripped off by agreeing in advance to an unknown price, eating omakase can be a pretty good deal, as long as you have a healthy sense of adventure. Several times I've ordered omakase and received larger portions for only half the price ordinary customers would have paid for the same dishes, not to mention special dishes that weren't even listed on the regular menu.