Like real pets, your Neopets will have to eat on a regular basis. Fortunately, Neopets can survive without being fed every single day, but feeding your Neopet a little something every day is a good way to keep him happy.
Every Neopet has a different interest in foods. Some will eat certain foods, others will not (Usuls DESPISE apples), and still others (like a Grarrl) will eat absolutely anything. If, for instance, your Grarrl hasn't been fed in two months, and you decide to explore the Neopets site for a little while, you may suddenly find that your Neopet has eaten half of the items in your inventory in order to stay alive! So, if you've been neglecting your Neopet, you better not put anything of value in your inventory!
If your Neopet's hunger level drops too low, it will not be able to fight in the Battledome. It's also more likely to become sad and, as a result, you'll come across more negative Random Events around the Neopets site than usual. The worst level of hunger is DYING, while the optimal one is SATIATED. If you feed your pet too much, however, it may become BLOATED, which some may argue is okay, but we would disagree because you wouldn't want your real life pets to become bloated, so why make your Neopet suffer?
Food can be found all over Neopia. Almost every world has at least one food store, and there are lots of stores around the Neopian Bazaar. For free food, you always visit the Giant Jelly or the Giant Omelette once per day. Another alternative is the Money Tree, where people donate items and NP to less fortunate users. The Money Tree is a great place to go if you NEED to, because going there greatly reduces your chances of getting a Random Event. Plus, certain locations like Coltzan's Shrine, the Fruit Machine, and the Tiki Tack Tombola may give out free food as well as other valuable items while playing. Or, if you have under 3,000 NP, you could always visit the Soup Faerie and feed your Neopet until it's full for free!
One final note - if you really care about your Neopet and want it to be taken care of, then when you go on vacation or can't play on the site for a while, it never hurts to go to the NeoLodge and check your Neopet in. For less than 16,000 NP, your Neopet can stay in the nicest room with all the luxuries for up to 28 days! Or, for a meager 140 NP, your Neopet can stay in barely livable conditions for 28 days... not great, but better than nothing.
Any foods with a rarity between 90 and 100 are considered "gourmet." The more you feed to your Neopet, the closer it is to making it onto the Gourmet Club High Scores Table! Feed your Neopet enough gourmet foods, and you may even get a trophy!