I'm currently designing a small form for a user create experience inside an app.
I'm wondering if it makes sense to have the last name field shown to the right of the first name OR down below the first name field.
Personally, I like to structure my form so that my fields are vertical. This leaves a clean area to the right to put validation messages, help icons, etc.
That said, you can have either way. No way is better than the other. As long as you have the Six Components of Web Forms, as defined in the "An Extensive Guide To Web Form Usability" article from Smashing Magazine, the your users should have no problems with your form.
The other thing you might think of is to group like fields into one area like Google does:
There's a Gestalt law of grouping saying that elements close to each other are perceived as belonging together: http://en.wikipedia.org/wiki/Principles_of_grouping#Proximity . I would even go as far and reckon that it was this perception that made you ask this question in the first place... ;-)
Thus in your example, "First Name" and "Username", and "Last name" and "Email" are grouped. I would thus recommend to switch the order to honor this. "First name" and "Last name" go together well, as do "Username" and "Email".
If that's against your intention, try to find way to group the horizontal elements closer or to follow one of the other Gestalt laws.