With due regards to Natalie Choate for initially putting together this list, below are the currently-legal methods for funding a Roth IRA account:

1. Contributions from compensation income. These are your regular annual contributions to the Roth IRA account. You are allowed (in 2019 and 2020) to contribute up to the lesser of your actual earned income compensation or $6,000 – provided your Modified Adjusted Gross Income (MAGI) is below the limits. The limit for 2019 is $193,000 if you are married filing jointly, or $122,000 if filing single or head of household. For 2020 the MAGI limits are a bit higher, $196,000 for MFJ and $124,000 for single and head of household filers.
2. Catch up contributions. If you are over age 50 during the tax year, you are eligible to make an additional contribution of $1,000 (for 2019 and 2020) above and beyond the “regular” contributions (#1 above). This figure would be reduced if your compensation income for the year is less than the total of the regular contribution plus the catch up contribution for the year.
3. Conversion from a traditional IRA. This is a special sort of taxable rollover from your traditional IRA account to a Roth IRA account. Ordinary income taxes are owed on the amount converted – but no penalty will be applied to all monies converted successfully from the traditional IRA to the Roth IRA.
4. Conversion from a Qualified Retirement Plan such as a 401(k). Much the same as #3 above, you may be eligible to convert funds from your QRP to a Roth IRA account. See this article for more details on how to accomplish this type of conversion.
5. Rollover from another Roth IRA. Ok, this one is a cheapy, but it represents another option. Rollovers from other Roth IRAs is always an acceptable option for funding a Roth IRA.
6. Rollover from a Roth 401(k). If you have a Roth 401(k) through your employer, within plan-specific limits you may be eligible to rollover the funds from the Roth 401(k) to your Roth IRA. Your own contributions to the account plus the growth can be rolled over tax-free to the Roth IRA account. There are no income limits on such conversions; however there can be limits on when you would have access to the rolled-over funds, depending upon your age at the time of the rollover, the age of the accounts, and other factors. See this article about Roth 401(k) distributions for more details.
7. Rollover from an inherited Qualified Retirement Plan (QRP). As detailed here, if you’ve inherited a Qualified Retirement Plan (such as a 401(k) plan) you may be eligible to convert those funds to a Roth IRA.
8. Failed rollovers or conversions. If you have attempted to make a rollover or a conversion into your Roth IRA and for some reason it is disallowed – such as you inadvertently had a higher MAGI than anticipated – the amount contributed to the account will be considered a regular contribution, subject to the following tax consequences:
   • a 6% excise tax per year for any excess contribution that is not removed from the account
   • the distributions from the traditional IRA or QRP (if a conversion) must be included in your gross income
   • the distributions will be subject to the additional 10% penalty for an early distribution not covered by an exception, successful conversion, or rollover
9. Certain military death benefits. As a result of the HEART Act, certain payments of Servicemember’s Group Life Insurance (SGLI) benefits can be rolled over into a with no tax or penalty. This article provides more details.
10. Qualified reservist distributions. A member of any of the US military reserves that has received compensation during active duty is eligible to place those funds in a Roth IRA at any time during the two years following active duty, without regard to normal contribution limits.
11. Exxon Valdez settlements. Certain individuals who received settlement compensation in connection with the Exxon Valdez oil spill can contribute up to $100,000 of the settlement into a Roth IRA without regard to normal contribution limits. Individuals include both the original plaintiffs and the heirs of the original plaintiffs. To be eligible for this contribution, the settlement would be claimed as income in the year received, and the contribution must be made during the same tax year as it is claimed as income.
12. Certain payments to employees of bankrupt airlines. Within 180 days of receipt of payments made in connection with bankruptcy of an airline, these payments or a portion of the payments, can be contributed to a Roth IRA without regard to regular limitations.

Note: #11 has some fairly narrow requirements; you can find more about these in IRS Publication 590. #12 has been eliminated and is no longer a valid option for funding a Roth IRA.