In what sense does forcing increase the width of a set-theoretic hierarchy? I've often read that, whereas top extensions increase the height of the cumulative hierarchy, forcing extensions increase its width (see, e.g., the answer to this question where's it's said that forcing extensions builds sideways instead of upwards; or see the last paragraph of John Burgess's review of a number of articles on the iterative conception). It has never been clear to me, though, what exactly is meant by "width" in this context. 
Is it "width" in the sense of the cardinality of a maximum antichain (see this Wiki article)? Or some other sense?    
 A: "Width" here is meant in the informal sense of how big each "level" $V_\alpha$ of the cumulative hierarchy is. The point is that, if $W[G]$ is a forcing extension of $W$, then they have the same number of levels, but each level of the extension is at least as large as each level of the original: $$(V_\alpha)^W\subseteq (V_\alpha)^{W[G]}.$$ Moreover, as soon as $\alpha$ is large enough (so that $G$ is $\mathbb{P}$-generic for some $\mathbb{P}\in (V_\alpha)^W$) then this containment is strict. (Assuming $W[G]\not=W$.)
Note that I'm not talking about the cardinality of the levels $(V_\alpha)^W$ vs. $(V_\alpha)^{W[G]}$ (since cardinality can change between $W$ and $W[G]$), I'm just comparing them directly as sets (in $W[G]$). And although frequently we'll have $W[G]\models \vert(V_\alpha)^{W}\vert<\vert (V_\alpha)^{W[G]}\vert$, this isn't always true: for instance, if we take $W[G]$ to be a forcing extension of $W$ by by adding a single Cohen real.
By the way, there's no sloppiness above: since $(V_\alpha)^W\in W$ we'll have $(V_\alpha)^W\in W[G]$, so the forcing extension can talk about levels of the ground model. Surprisingly, even more is true - $W$ will be definable in $W[G]$, by work of Woodin and Laver - but that's not needed here.

The orthogonal notion is that of a top extension: $W_1$ is a top extension of $W_2$ if (1) $W_1$ is an end extension of $W_2$ (that is, $a\in W_2, W_1\models b\in a \implies b\in W_2$) - note that this implies that the ordinals of $W_2$ are ordinals in $W_1$ as well - and (2) for each $\alpha\in ON^{W_2}$ we have $(V_\alpha)^{W_2}=(V_\alpha)^{W_1}$. So a top extension only makes the universe taller.
For example, if $W\models \alpha$ is inaccessible, then $W$ is a top extension of $(V_\alpha)^W$.
