Sometimes it can be seen with artifacts appearing on screen or in games it can look like tears appearing. There is also a snowy effect or lack of colour and strange objects appearing. They can appear at both startup (usually gobblygook writing during POST) and when in use.
There are programmes you can run to test your card for both Nvidia and ATI/AMD but they do put a lot of strain on the card and may make it fail altogether. DirectX has a small test built in to it. Use Start > Run > dxdiag and go to the Video tab and run the tests.