When soneone comes around to get you registered to vote, is that the point at which you elect to become a registered Republican or Democrat ?
In Ohio, yes. You’d check a box on the form stating that you’re registering as a Democrat, Republican or Independent. If you choose to vote in another party’s primary the next time around, however, your registration is automatically changed to that party’s. You could change every time, if you wanted to, but most people don’t.
Other states’ voter-registration policies vary.
Some states have non-partisan voter registration. North Dakota has no voter registration. When I lived in Iowa, they had partisan registration but you could register or re-register at the polling place, on election day.
To reiterate what others are saying, it’s a state-by-state thing. I was registered to vote in Virginia when I transferred my driver’s license, and they didn’t even ask me for a political affiliation – with an open primary where you pick which one to vote in they don’t really matter except for political junk mail.
In states with closed primaries, political affiliation is important because the only way to vote in (say) the Republican primary is to be registered as a Republican. But since changing affiliation is much simpler than verifying that someone exists, many states let you change your affiliation at the polls, making the primaries effectively open.
Most states will offer to register you at the DMV when you apply for/renew your driver’s license or state ID card. You can also do it at various government offices and sometimes at the local library.
And people can go around and have people fill out registration cards, and in most states they’re required to turn in any and all filled-out forms they collect.
Finally, there are other political parties you can sign up for. Back in Pittsburgh, I was registered as a Libertarian for several years, until I realized that the only way to truly be able to vote for the mayor was to be a Democrat and vote in the primary. (Pgh is heavily Democratic, and the Republicans only offer up candidates for local offices to keep up the show.)
And apparently none of this applies to North Dakota, where all the locals apparently just get together at the stoplight for a show of hands 
That’s a little disingenuous for GQ. We all know that North Dakota does not have any stoplights.
This varies from state to state. Ohio, for example, has no true registration by party. All there is is a notation as to which party’s primary you voted in last. You are free to request any party’s ballot the next time you vote in a primary, regardless of this notation.
You can see it on the CT registration (PDF) in section 9:
http://www.ct.gov/sots/LIB/sots/ElectionServices/ElectForms/electforms/ed671.pdf
In New York, you can choose from any of the state parties that are on the ballot: Democratic, Republican, Independence, Conservative, or Working Families, or as an independent. You can thus vote in primaries for the party you’ve chosen (independents hold no primaries). You need to reregister to change party affiliation, so no last-minute changes.
Why so many parties? You get on the ballot if your party gets 50,000 votes in a gubenatorial election. Candidates can count votes for them on multiple party lines, so if you’re running as both a Democratic and a Working Families candidate (a common pairing), you get votes from either line. Thus if a Republican can’t bring himself to vote Democratic, he can vote Working Familys. The Republican and Conservative parties are also usually linked (except in Schenectady County, where the Conservative party is a front for the police and fireman’s unions). Independence endorses both sides, but tends to lean Democratic these days.
You can vote for primary candidates for the party you’re registered in. That’s how the police and firemen took over the Conservative party here – they joined, elected their own candidates and use the power of being able to offer an extra ballot line as a way to get candidates to do what the unions wanted. Small party primaries are rare since they don’t have a lot of members, so no one mounts a challenge. I recall one person talking about voting in the Conservative primary and discovering he was one of only two eligible voters in that district – and the other wasn’t showing up. No secret ballot there.
There are also other parties that put forth candidates on the fringe.
In Colorado, there’s a spot on the registration form to register as a Democrat, Republican, or Independent. You have to register with a party affiliation to vote in a primary election or caucus.
When I lived in Michigan, there was no party registration, but you could only vote in one or the other primary.
In Arizona, you do register by party. You can vote in your own party’s primary, but are free to vote for any party in the general election.