What is the definition of a succeeding tax year?

It means the tax year that immediately follows.

Individuals are calendar year filers, so here's an example: You have a tax credit on your 2010 tax return that you couldn't fully use because it was more than your tax liability, but it can by carried forward to the next succeeding tax year. Therefore you can claim the unused amount as a tax credit on your 2011 tax return.

Tax before credits = $500

Nonrefundable business credit = $800

Tax = $0

$300 of the credit wasn't used & it's a credit you're allowed to carry forward, then you can use the remaining $300 of unused credit in the next year.

There is usually a limit to how many years you can carry a credit forward before it is lost (unusable). If you have several years of business losses, you may end up with tax credits you never use.