One thing to add to Kim's excellent response: if your adjusted gross income is too high (depending on your tax filing status), you may be ineligible to contribute to a Roth IRA. The income range varies from year to year, but for 2012, for instance, it's $110,000-$125,000 for singles, or $173,000-$183,000 for married couples. So if you're married, for instance, and your adjusted gross income (bottom line number on the 1st page of your 1040) is more than $183,000, you can't contribute to a Roth IRA.
But you certainly could contribute to a non-deductible traditional IRA, and then convert that IRA to a Roth IRA. There's no income limit on Roth conversions. If this IRA would be the only one you have, it's a pretty simple process. If you have existing traditional deductible IRAs, then the conversion process gets a little messier and you should really pull in a good tax advisor, to make sure the conversion is done properly.
Hi - there are no constraints on participating in a company 401k or contributing to an IRA (Roth or Traditional) as a result of receiving Social Security retirement benefits. The key requirement for contributing to retirement plans is that you must have earned income - wages or salary or self-employment income. Have you reached your Full Retirement Age (FRA) for Social Security? If you have, then there is no "give-back" on your Social Security benefits. If you haven't, then you may lose some of your Social Security benefits depending on how much income you earn. If earned income is less than $14,640 in 2012, then there is no give back. Social Security takes back $1 in benefits for every $2 of earned income that exceeds $14,640. For details, see www.ssa.gov - look in the FAQs for "How work affects Social Security retirement payments". Your social Security benefits may also be subject to income tax depending on your total income although no more than 85% of your benefits will be taxable. Look in the FAQs for "Paying income tax on Social Security benefits". Good luck!
Excellent and thorough answers. I would only add that even though your contributions can continue - you are still subject to minimum distributions the year after you turn 70 1/2. ( Except if it is a Roth) Hence if you are earning income well into your 70's or 80's it may become a " revolving door" at some point... put in , take out, put in, take out and so on. A thoughtful analysis might reveal that it's not worth contributing to a traditional plan, even if you are eligible. Of course it would depend on your overall situation and goals. Best of luck.