Let’s Talk Birds originally wrote the following article about Cockatiels, but generally this advice 
can apply to most pet birds and parrots (source: LTB Cockatiel e-book)
Sad Times
The death of one of a pair of cockatiels is good reason for concern on your part. Your remaining cockatiel will mourn the loss of its companion and you will need to give your bird the extra attention it needs to help it through this time. You will also need to monitor your cockatiel closely to check that it stays healthy – monitor its eating, drinking, sleeping and the droppings.
All cockatiels react differently to this situation so there are no hard and fast rules. The amount of time the two birds were together and whether or not they had bred and successfully reared chicks does impact on the situation.
Some cockatiels do fret terribly with the loss of their companion yet others come round fairly quickly. They do go through a slight depression and you’ll have to be careful that the loss of its friend doesn’t turn your cockatiel into a screamer.
Do not consider obtaining another cockatiel, especially not in the early stages. Introducing a new cockatiel at this time normally does more harm than good. Watch your bird closely and see how it gets along in the next few weeks.
I know of someone who had a male cockatiel die, leaving behind his female companion of ten years. The female cockatiel cried for a while and turned to its owner for more attention than usual. The male cockatiel wasn’t replaced, and the female coped just fine, taking a week or two to adjust to the situation. These two cockatiels, however, did live in separate cages side-by-side and had never bred.