How does trademark relate to acquiring domain names?

Say I'm able to trademark the word "widgetville" for an online game that I'm building.

But the domain "widgetville.com" (and pretty much every major TLD of the word) are all registered, but just squatted on...no actual content.

Do I have the ability, legally, to "force" the owners of those domains to hand 'em over since they're using a trademarked word?

Generally, no, because trademarks only offer protections in specific business sectors a.k.a. classes and real protection arises only from demonstrated continuous use of the name in the applicable sector. Unless your mark was famous or well-known prior to the date the domain name was registered, you will not have a solid case for a domain name dispute especially if you received your trademark registration after the domain name was registered. Either way, you'd have to invest time and legal fees to put pressure on the domain owners.

Note that the verb "trademark" actually involves using the name in business. Registering a trademark does actually not offer any legal protection in itself. Registration serves as a deterrent against others who may try to use the name after you do and sometimes can serve as supplemental measure in legal contexts.

There are two great blogs on domain names and trademarks:

howardneu.com a lawyer and pioneer in Internet domain lawsuits.

TheDomains.com Blogger and owner of thousands of domain names, expert in lawsuits etc.

Check out those blogs.
My understanding is that if you take a trademark after someone else took the domain name you will have very little chance to win a UDRP to get the domain.

My advice to you: try to buy the domain name, preferably using escrow services like escrow.com or Sedo.com. If the domain is listed for sale on Sedo.com you can also negotiate there without revealing who you are before the end of the negotiation which can be advantageous if a google search on your name would reveal that you need the domain (say someone finds out you start a service on the subject as opposed to you being just a domain name flipper).

Like The NeoTycoon says "generaly - no".
I have looked at some cases when somebody tries to obtain domain in legal action. It is very difficult to win this battle, especially if the domain owner do not use the domain (parked domain) or use it a way that do not violate you trade mark.

There is something called Reverse domain hijacking. You can look at this page about some UDRP legal cases.

Trademarking doesn't give you much rights. (In fact, it usually doesn't protect the use of your trademark of others either. It's more of a "we use this name for that"-seal). However, in some cases, operating a business in the same name may. Even after the fact.

When Groupon launched in the UK, their name was squatted. The story goes that they first offered the squatter some money. The squatter, knowing Groupon was successful asked for a higher price. Groupon then incorporated in the UK, and appealed to have the domain taken over. Apparently they won. ICANN frowns upon squatters, and if you do it right, you MAY be in luck. Of course, if you're not already in business, making lots of money, you're less likely to be in luck.