A 401(k) must be sponsored by an employer, and the only way to make contributions is via payroll deduction.
An IRA is a private account you set up with a bank, stock broker or investment advisor.
A Roth IRA and a "Roth" 401(k) (a post-tax 401(k)) are similar in that you contribute money after taxes--you pay taxes on the money now and don't take a tax deduction for contributions. When you retire, you withdraw the money tax-free.
However, a Roth IRA and a post-tax 401(k) have different contribution rules, different contribution limits, and different withdrawal rules.
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