I am not eligible for Traditional IRA this year (2020). But I am still under the limits for Roth IRA. I will contribute to Roth IRA in full. I heard that there is a Non-deductible IRA. Can I do the Non-deductible IRA as well? That is Can I contribute to Roth IRA and Non-deductible IRA at the same time?
No, if you max out the Roth you cannot also contribute to a Traditional IRA, neither pretax nor after-tax. The max amount you can contribute to an IRA is the total of all contributions you make to any kind of IRA. In 2021, the total contribution limit is $6000 if you are under age 50, and $7000 if you are 50 or older. You could theoretically contribute less than the limit to the Roth, and then you could also make after-tax contributions to a Traditional IRA as long as the total is under the contribution limit. However, you would rarely ever want to put after-tax money into a Traditional IRA while you still have the option to put it into a Roth.
In the future, if you are no longer eligible to contribute to a Roth due to income limits, then you can make after-tax contributions to a Traditional IRA, and if you wish, convert them to a Roth IRA without penalty. (Usually you would aim to convert as soon as possible if you didn't already have a lot of other money in Traditional IRAs.)
Side note: the cutoff for contributing to an IRA for 2020 is April 15, 2021. This means between now and April 15, 2021 you can actually contribute up to $12,000/$14,000 to the Roth. You can assign $6/$7K to 2020 and $6/$7K to 2021. After that you won't be able to contribute again until Jan 1, 2022.
Yes, but no...
The core point is that the total of the two is limited to 6000 (or 7000 if older than 50). Within this limit, you can spread your contributions between any number of IRAs and Roth IRAs.
Also: You can always contribute to an IRA, there is no upper income limit. The income limit is for being able to deduct the contribution from your taxable income. However, and that’s probably what you meant, it is not that useful once you are over the limit; better to contribute to the Roth IRA instead.
Furthermore, once you get over the Roth IRA income limit, you can go back to contributing to the IRA and convert it to Roth IRA right away - as the contribution is from after-tax money, there is no tax impact, and you get the money into the Roth IRA for free (this is called a ‘backdoor contribution’).
However, if you already have pre-tax money in your IRA (from previous years), make sure you understand the IRS view to this - they consider those contributions ‘merged’, no matter how you handle it, which results in complex consequences and paperwork.