Photo from nextstop
In a sushi restaurant, omakase means "trust the chef". Instead of ordering off a menu, you let the chef decide what he wants to give you. The pleasure of omakase doesn't come cheap, but it's usually worth the splurge. In L.A., you can go from the extravagant ($350 at Urasawa) to the budget friendly ($20 at SugarFish and $40 at Kanpachi).
The best Omakase meals I have tried in L.A. are at Hiko Sushi (~$60 minimum), Echigo ($70), and the best of them all, Sushi Zo in Culver City ($85 min).
Only serious sushi restaurants serve omakase right, and with that in mind, be prepared for some very serious sushi chefs behind the counter. They may throw you out if your cell phone rings (the chef at Hiko is infamous for this), or they may glare at you if you dip your sushi the wrong way into the soy sauce. Study up on sushi dining etiquette, and be sure to read any signs they have posted in the restaurant about this subject. Also, be prepared for weirdness. At Sushi Zo, I had my first taste of codfish milt (fish sperm! it was actually good). Speak up if you have a low tolerance for the bizarre.