Do you want to print and play other people's games, or find a site that will manufacture your board game? Your question is really unclear.
For the former, you'll want to search for "print & play board games," and for the latter, you'll want to search for "board games print on demand." The Game Crafter is a popular option.
Yes, we ship throughout the world via USPS and UPS.
If you're in Europe, especially if you're going to make a bunch of sets of these, you might take a look at shipping costs and whether there's some comparable service in Europe that might cost less after shipping. I'd expect that shipping from the US is going to add cost.
EDIT: I'd also add -- and this is from an American, albeit not one who plays a lot of board games -- that my impression is that Europe has a stronger boardgame tradition than the US. I'd think that the people making them probably have some kind of domestic industry.
EDIT2: I have no familiarity with any of these companies, but if you just want names and not familiarity, they're out there:
I've had this list of print-and-play games from Randomskill (containing games that are not of Randomskill as well) bookmarked for a while, and haven't actually tried any of them...because I don't have a printer. I have played a few of these games in their commercial versions though, but not the free ones.