TITLE: What is the symmetric monoidal structure on the $(\infty,1)$-category of spectra?
QUESTION [11 upvotes]: The $(\infty, 1)$ category $Sp$ of spectra as defined by Lurie in Higher Algebra has the structure of a symmetric monoidal category. Although I know the definition of symmetric monoidal category in the $(\infty, 1)$ setting and can reasonably follow Lurie's arguments in Higher Algebra as to why $Sp$ has such a structure, I don't understand it well enough to think about it intuitively.
So my question is, what does the symmetric monoidal structure on $Sp$ look like? How is this related to the symmetric monoidal structure on symmetric or orthogonal spectra in the ordinary categorical setting? How may I picture ring spectra and other such objects arising from the monoidal structure?

REPLY [6 votes]: I thought it might be worth mentioning, as an addendum to the other nice answers, that there is also an $\infty$-categorical approach to the smash product that looks more like the traditional constructions: namely, you can just redo the most classical construction of the smash product  on the stable homotopy category (as in Adams's book) using $\infty$-categories. (I suppose this might count as a "folk theorem", and unfortunately I don't think it's been written down in detail anywhere; I learned the idea from Justin Noel.)
Start with the $\infty$-category $\mathrm{PSp}$ of "prespectra", meaning sequences of pointed spaces $X_n$ with maps $\Sigma X_n \to X_{n+1}$. Using the smash product of pointed spaces you can enhance this to an $\infty$-operad: a multimorphism $(X, Y) \to Z$ is given by maps $X_n \wedge Y_m \to Z_{n+m}$ compatible with the structure maps in $X,Y,Z$ (and similarly for multimorphisms with more inputs). 
Now consider the subcategory $\mathrm{Sp}$ of "spectra" (or "$\Omega$-spectra"), meaning prespectra $X$ where the adjoint maps $X_n \to \Omega X_{n+1}$ are equivalences. If you restrict the operad structure to this subcategory, I claim you get a symmetric monoidal $\infty$-category: for spectra $X$, $Y$, $Z$, multimorphisms $(X,Y) \to Z$ are represented by maps $X \wedge Y \to Z$ for a spectrum $X \wedge Y$. This spectrum you can define by choosing sequences $i_n$, $j_n$ with $i_n+j_n = n$ (with both unbounded) and setting $(X \wedge Y)_n := X_{i_n} \wedge Y_{j_n}$ - what Adams calls a "handicrafted smash product", if I recall correctly.
Actually working this out precisely would take a fair bit of (probably quite annoying) work, however, and in terms of applications I doubt it would have any advantages over the constructions Dylan and Denis describe.