This is similar to another question on here, but my concern is slightly different.
I've been developing a web app for the past 2 years and registered the domain name for it about a year ago. Other than that, I have no claim to having come up with the name (I have not trademarked it). Another business has taken almost the exact same domain name (they've put an 's' on the end), and launched a site that targets pretty much the same service I'm launching, though I haven't launched it yet. So basically:
I registered this site first
They registered the site second with an 's' tacked on the end (liquidgel.com vs liquidgels.com, for example)
They are based in Denmark, I am based in the US, but we both target the same demographic.
I have not trademarked the name though USPTO.
Is it still possible for me to use the name? I'd really like to, even though it would more than likely cause confusion.
This is a complicated question but I'll try to flesh out an answer. In summary, start using your mark ASAP. The worst thing that happens is that you get a C&D letter from the Danish company and then you either fight or change your name.
Trademark rights are quite different in the US and Europe. In the US, you must use a mark in commerce to get any rights at all. Registering a domain name does not count. If you have not previously used the mark in commerce, then the Danish company might be able to stop you. If you have, then the Danish company probably can't stop you in the US but may be able to stop you in Europe. Maybe you don't care about that.
The good thing about US law is that you have trademark rights even if you don't register. Registering does provide benefits as outlined here. It is up to you whether it is worth the money to pay for a registration (assuming you can get one).
In Europe, you get trademark rights by registering. This is generally not worth the effort for small companies.
Trademark the name, if you can. Investigate if any trademark exists or is pending. It's inexpensive but can take a while to secure so start now.
Get some content up on your site to show active usage of the domain name. A landing page will be enough for now but more would be better. This will help if they mount a challenge on your ownership of the name.