What does ‘Put one's big boy (girl) pants on’ mean?

In the United States, toilet-training a child typically starts with diapers. An intermediate step is "trainers" or "pull-ups", which are basically diapers with elastic: the kid gets used to the idea that s/he shouldn't just "let go" at any moment, but the absorption is there so that it's not a total disaster if it happens.

Finally, when the kid has demonstrated his/her self-control, the much-anticipated "big girl pants" or "big boy pants" are awarded: the first pair of underwear (and whatever clothes go on top) that don't humiliate the kid.

"Put on your big boy/girl pant/ies" is a playful but rather insulting way of saying that, up till now, you've been acting like a child who hasn't been potty-trained yet: time to grow up and act like an adult. This is definitely a slang usage; used among friends in an obviously-joking way it probably won't cause offense, but in any other context it could be quite offensive.

Your second example (Put pants on before you "hangout" with President Obama on Google+) is in a completely different context, and is meant to be funny but not insulting. Google+ hangouts are multi-person video chats; the writer is referring to the fact that many people these days dress extremely casually (or don't dress at all!) when chatting by video; if you're going to chat with the President of the United States, it would be best to put on some pants first. (To hang out is a slang phrase meaning "to socialize in an informal setting", a hangout is a place where people hang out. In this article, there's an implied, slightly risqué third meaning: a certain part of your anatomy might "hang out" if you're not wearing pants.)

Finally, a note on American/British usage:

• in American English, pants are synonymous with trousers, while panties are girls' or women's underwear.
• in British usage, pants are always underwear - men's or women's. (Edit: @BrianNixon informs me that "pants" is more likely to be men's underwear; women's are more usually called knickers.
• the Brits also use pants as a negative slang adjective, in much the same way that an American would say crap. (In other words, it's a "dirty" word - little kids shouldn't use it - but it's not very offensive, and it's used for humorous effect.) A person who's just cooked a meal for a friend - and has only just realized that it came out badly - might say "This is pants, isn't it?" by way of apology.
• Mad as pants is British slang for "crazy", either a person or an idea.

The inclusion of a modifier like "big boy/girl" is crucial to the meaning: stop behaving childishly.

The expression alludes to literally telling a child to put on more mature clothes, "big boy/girl" being used in place of "mature" for ease of comprehension. One might literally tell a child to "put on their big boy/girl pants" before a formal event. By association, the instruction also implies that one should act more maturely.

As used in the referenced quote, the speaker is deliberately condescending, treating the Romney campaign as children and telling them to act less childishly.

It basically means that the person should man up, step up, or take responsibility depending on context. You'll also see it as panties rather than pants, as in 'put on your big girl panties and (do whatever)'.

In this case, the author soon follows with the suggestion that Romney's campaign defend his record.

It was common practice in the beginning of the twentieth century (and perhaps even earier) to dress young boys mostly in short, or knee pants. When a child was considered more mature, he got his "first pair of long pants," hence, big boy pants.

See, e.g.:

http://en.wikipedia.org/wiki/Long_Pants

Or so my 91 year old father tells me.