乳製品, にゅうせいひん (nyu-sei-hin) is the general term for dairy products. If you're eating out or shopping in a grocery store, this word will cover you for all dairy products that might be used as ingredients in cooking.
Pro Tip: Look out for any words on the packaging that includes the character 乳 (nyu). This kanji refers to milk products. It's also used in soy milk (豆乳) but it generally refers to a dairy form of milk. If in doubt, point it out to someone, using the phrase nyu-sei-hin to double check if it is a dairy product.
Other Dairy Products
Egg - 卵, たまご (tamago)
Butter - バター (bataa)
Cream - クリーム (kuri-mu)
Cheese - チーズ (chi-zu)
Cow's Milk - 牛乳, ぎゅうにゅう (gyu-nyu)
お肉, おにく (oniku) is the general term for meat products. You might see the shorter form 肉 (niku) used as well when looking for ingredients.
Pro Tip 1 - If the kanji 肉 is combined with another unfamiliar kanji, the likelihood is that the ingredient derives from an animal.
Pro Tip 2 - Also look for the word エキス which means "extract." A lot of food in Japan that might seem like it's vegan-friendly may have meat or fish extracts as an ingredient. Look out for the combination of 肉 and エキス.
Other Meat Products
Beef - 牛肉, ぎゅうにく (gyu-niku)
Pork - 豚肉, ぶたにく (buta-niku)
Chicken - 鶏肉, とりにく (tori-niku)
Goat - ヤギ, やぎ (yagi)
Horse - 馬肉, ばにく (ba-niku)
Lamb - ラム肉, らむにく (ramu-niku)
Duck - 家鴨肉, あひるにく (ahiru-niku)
シーフード (shi-fu-do) is a large part of the Japanese diet. Different types of fish and shellfish are available everywhere in Japan, and a lot of it makes it as an ingredient in the most unusual places. Do look out for seafood-related kanji when you are doing your weekly shop.
Pro Tip - Pay special attention to だし (dashi) or stock. Dashi in its usual form is a soup base that uses seafood extracts to give it its signature flavor. The best vegan alternative is to look for a konbu dashi (昆布だし) which is a seaweed-based stock.
Pro Tip 2 - Like we mentioned in the above section, keep an eye out for エキス as a lot of food in Japan contains seafood extracts.
Other Seafood Products
Fish - 魚, さかな (sakana) (Click here for a list of different types of fish)
Shellfish - 貝, かい (kai)
Shrimp - エビ or えび (ebi)
Salmon Roe - いくら (ikura)
Below are some other animal products that you should look out for if you're a vegan.
Honey - 蜂蜜, はちみつ (hachimitsu)
Gelatin - ゼラチン (zerachin)
Leather - 革, かわ (kawa) or レザー (rezaa)