ETA advice from the U.S

Yes, you do need it if you are a US citizen.

Only apply at the official UK government website Gov.UK or the official apps from the Apple store or the Google store.

Here is the link to apply along with links to the official apps:
Apply for an ETA

The current cost is £20 and is valid for 2 years from date of approval.

I would add that you shouldn't leave it until the last minute as there have been some technical glitches at times.

Also, if you happen to be flying IcelandAir, they'll want to see your email acknowledgment even though the instructions say it is "electronically attached to your passport and you can delete the email".