I love Asian food, especially Thai. Many of my recipes are Asian inspired and use a myriad of Asian ingredients. Almost everything listed is widely available at most large grocery chains. Save on Foods, IGA and Superstore all have great ethnic food sections. Safeway is less stocked than most. If you have an Asian supermarket in your area (like T&T), make use of it as you will find everything you need and it will cost less too!
Fish Sauce is a staple of Thai & Vietnamese cooking. It is derived from fish and salt that has been left to ferment for several months. Fish sauce is used in sauces, marinades and as a seasoning condiment (like we use salt). To quote Thai Food & Travel, Fish sauce "is indispensable in the Thai kitchen as Thai food wouldn't be quite the same without it".
Garlic Chili Sauce is a Vietnamese chili sauce similar to Sriracha and Sambal (see below). The main difference is that Garlic Chili Sauce is hotter and runnier than Sriracha and Sambal. Generally, they can be used interchangeably.
Lemongrass is an herb widely used in Asian cooking and native to India. It can be dried, powdered or used fresh. It imparts a citrus flavour and is often used in marinades and to flavour soups. See here about choosing your lemon grass and how to prepare it for cooking.
Lime Leaves are native to South Asia and are common in Thai cooking. The leaves can be used fresh or dried and the fresh leaves can be frozen for future use. Like lemongrass, lime leaves are most often used in marinades and soups.
Mirin is a sweet Japanese cooking wine with a low alcohol content. It is used to make teriyaki and other sauces.
Oyster Sauce is common in Chinese cooking. It is a thick, savory sauce made from oyster extracts.
Red Thai Chili Pepper, also known as "birds eye chili", are small, thin, hot peppers. They are about an inch long and usually sold in small bags with a couple dozen or more pieces.
Sake or Cooking Sake, is a Japanese alcoholic beverage made from rice. It is often called rice wine, although unlike wine, it is brewed not fermented.
Sambal or Sambal Oelek is a hot chili sauce made from several types of chili peppers. It is common in Indonesia, Malaysia, and the Philippines.
Shirataki Noodles are gluten free, low carbohydrate, low calorie, thin, translucent noodles made from yam starch. In North America the most common types are made from a blend of tofu and yam starch. The noodles have little to no flavour and tend to take on the flavours of their cooking sauce or liquid. Shirataki noodles are incredibly versatile and can be used in place of traditional noodles in most all dishes.
Soy Sauce is a Chinese cooking staple made from fermented soy beans. It also contains wheat so if you have any wheat or gluten intolerance it's important to use gluten free soy sauce.
Sriracha is a Thai chili sauce used to flavour sauces, marinades and as a condiment. The main ingredients are chili peppers, galic, vinegar, salt and sugar.
Thai Curry Pastes are a moist blend of fresh ground herbs and spices. Read more about Thai Curry and how it is different from Indian Curry here. The common varieties are red, yellow and green.
Other staples of Asian cooking include fresh garlic and ginger, rice wine vinegar, peanut oil, coconut milk, cilantro (and corrinader, the latter referring to the seed and the former refering to the leaf) and several ground herbs and spices (turmeric, tamarind, cumin, star anise and galangal, for example).
If you enjoy Asian cooking, I recommend stocking your pantry (and fridge) with good quality soy sauce (gluten free if needed), fish sauce, a chili sauce, red & green chili pastes, a can or two of coconut milk, rice wine vinegar and a jar each of minced ginger and minced garlic. Those will get you started. Later on you can add more to your Asian pantry as needed.
A well stocked pantry leads to good cooking!