Upvote:6
If you are a citizen of a visa waiver program (VWP) country, you can apply for ESTA. Answer all the questions truthfully. If it is granted, you can travel to the US. If it is refused, you will need to have a B-2 visa to travel to the US, for which you will have to apply at the nearest US embassy or consulate.
The ESTA application could be refused because of the prior visa refusal, and it should be refused if your first stay in the US was authorized under the VWP and exceeded the allowed 90 days, because that would make you ineligible to use the visa waiver program.
A visa refusal 20 years ago will probably not pose a huge problem for a visa application today, as your circumstances will be very different. The refusal is likely to have been because you did not present a convincing case that you would be able to spend one year in the US without working or otherwise violating the conditions of admission as a visitor. One-year visits are unusual, so the refusal is not surprising. The evaluation of a visa application for "a holiday with my family" will be rather different.