Food that was thawed in the refrigerator can safely be refrozen without cooking, as long as it has not been above 40°F (4°C) for more than two hours. This applies to raw meats, poultry, fish, vegetables, and most other foods. The food is safe because it never entered the danger zone long enough for significant bacterial growth.
Food that was thawed using cold water or the microwave should be cooked before refreezing. These methods allow portions of the food to enter the danger zone during the thawing process, so cooking is necessary to kill any bacteria that may have begun growing.
Food that was left out on the counter at room temperature for more than two hours should not be refrozen (or eaten). Discard it.
While refreezing is often safe, it does come with a quality cost. Each freeze-thaw cycle causes additional ice crystal damage to the food's cells, resulting in more moisture loss, softer texture, and a greater chance of freezer burn. Refrozen meat may be noticeably drier when cooked, and refrozen fruits will be mushier.
To minimize quality loss when refreezing, package the food in fresh, airtight packaging (not the same bag it thawed in, which may have lost its seal), freeze as quickly as possible, and plan to use refrozen items in cooked dishes where texture changes are less noticeable (soups, stews, casseroles, sauces).
A better approach than refreezing raw food is to cook it first and then freeze the cooked result. Cooking kills bacteria, changes the food's structure in a controlled way, and often produces a product that freezes well in its cooked form. For example, raw chicken thawed in the fridge can be cooked into a stir-fry, soup, or casserole and then frozen as a prepared meal.
This is especially useful when you thaw more food than you need, or when plans change and you will not be using the thawed food in time. Cook it, cool it, package it properly, and freeze it — you now have a ready-made meal.
After a power outage, food in the freezer may have partially or fully thawed. If the food still contains ice crystals and feels cold to the touch (at or below 40°F / 4°C), it can safely be refrozen. Quality will suffer somewhat, but the food is safe.
If the food has fully thawed and been above 40°F for more than two hours, it should be discarded — with a few exceptions. Hard cheeses, bread, plain muffins, fruit without added sugar, and plain vegetables are lower-risk items that may be safe even after extended thawing, though they should be inspected for off smells or unusual texture before use.