Pufferfish have a lot of personality for a fish. They will beg for food from their owners. At the same time, they have have some specialized requirements for their food. Many household foods, particularly seafood make great pufferfish fare.
Hard, crunch food like unshelled shrimp help maintain a puffer's teeth. On the other hand, crabs and shrimp contain a chemical called thiaminase, which interferes with the absorption of vitamin B1. You can get around this by soaking thawed crab and shrimp in B1 supplements, available at high-end pet shops and online, or only feeding these sparingly.
A puffer will eagerly eat clams, cockles and mussels. Mussels probably make the best food for puffers, since mussels have thinner shells that puffers can crunch with their beaks. You can feed these items whole to larger puffers, or finely dice them for smaller species. Like crustaceans, mussels contain thaminase, but you can soak them in supplements to make up for it. A varied diet can overcome this issue as well.
Pufferfish also love frozen and fresh fish. You can take frozen fish and shave off chunks with a cheese grater. Avoid oily species like herring and tuna. Salmon makes provides good nutrition since it is rich in caratinoids, which help maintain color in aquarium fish. Keep in mind that puffers need crunch foods in their diet as well, so they cannot live on fish alone.
Grocery store squid also makes a good puffer food. You many need to chop it to size, though the cheese-grater trick works well on frozen squid. Like fish, it has the same drawback; no crunch parts. Since puffers need hard foods to wear their beaks down, they cannot live on a steady diet of squid.