I put $5,500 in my Roth, and learned that I'm not eligible to (I make too much money). What's going to happen to me, come the time of reckoning (tax time), and what can I do about this before then?
If you don't take out your excess contribution (including earnings) by 6 months after your tax filing deadline (i.e. October 15 of the year after the year the contribution was designated under), you will pay a 6% penalty every year until you take it out or until it can count under the unused contribution limit for that year.
One thing you can do (again, before October 15 of the year after the year the contribution was designated under) is re-characterize the Roth IRA contribution into a Traditional IRA contribution. Traditional IRA contributions do not have income limits (there are income limits to deduct a Traditional IRA contribution if you are covered by an employer plan, but that is another matter). Then, assuming you didn't have any other money in pre-tax IRAs, you can convert the money in that Traditional IRA into Roth IRA. You would have to pay taxes on the earnings between the contribution and the conversion, but that probably won't be much. This is a "backdoor" Roth IRA contribution -- the result is basically the same as a regular Roth IRA contribution, but with no income limits.