The most basic difference is that the Roth contributions are made after-tax while the Traditional IRA contributions are usually deductible from income. Both Roth and Traditional IRAs provide for tax-deferred growth of your assets.
However, the contributions you make into your IRA are pretax (or, more accurately, tax-deductible), and contributions you make to your Roth IRA are after-tax. The annual contribution limits for each are usually the same, and, in fact, a person can contribute up to this amount in either of these combined in a given year.
This is assuming someone is eligible to contribute. For Roth IRAs, eligibility is not affected if you or your spouse has a qualified plan at work. The only thing that limits the ability to contribute to a Roth IRA are income limits, which are updated every year.
In a Traditional IRA, the ability to deduct the contributions will be affected if either spouse has a qualified plan at work, and, if so, the individual or couple would have to be under income limits in order to deduct their contributions to the Traditional IRA.
It is possible to contribute to a Traditional IRA without being able to deduct the contributions, and this is known as a non-deductible IRA: it will still grow tax deferred, and the withdrawals will only be partially taxable until the cost basis has been used up.
There is such thing as a Back-Door Roth, which eliminates the income limitation by allowing people to treat contributions as conversions from non-deductible IRAs.