An Eat Mission is where you go to try a dish at your neighbor's cafe, sort of like a Special of the Day at their cafe. Sometimes in order to complete a goal, you have to complete a certain amount of Eat Missions before you complete the task or complete the goal. 'Well how do I complete an Eat Mission' you ask? Well it's actually really simple. You can complete an Eat Mission by either:
1. Going to your neighbor bar at the bottom of your screen, scroll over your neighbor's pic, and click on the option that says 'Today's Dish'. It looks like this: Once you click on 'Today's Dish,' a pop-up will, well, pop-up and it will say, 'Hey [your name here]! Would you help me by trying my daily special?' The dish is generated randomly. Click on the 'Yes' button at the bottom of the pop-up. It will have a little show of the dish being eaten until there's nothing left. Once that's done, the pop-up will say, 'Thanks! Here's a reward for helping me out...' You will get Cafe Points and Coins. Depending on your level, the number of Cafe Points and Coins you receive will vary. Now there are 2 buttons: A 'Share' button that allows you to share the 'Today's Dish' with your neighbor by posting to their Wall, and there's the 'X' button, where you can just close the pop-up without sharing.
2. Going to your neighbor bar at the bottom of your screen, scroll over your neighbor's pic, and click on the option that says 'Visit'. It looks like this: This will take you to your neighbor's cafe. Once there, you can spice your neighbor's stove and complete an Eat Mission. Now you can either scroll over your neighbor's pic and click on 'Today's Dish', or on the right-hand side of your screen, there will a blue man holding a fork and spoon like this: The same pop-up will come up and the steps are the same like in the first option.
There! Now you know how to complete an Eat Mission and what it is. Hope this helped! :D