What is DAG and what has it to do with the ideas of Voevodsky? In Toen's and Vezzosi's article From HAG to DAG: derived moduli stacks a kind of definition of DAG is given. I am not an expert and can't see what's the relation between DAG and the motivic cohomology ideas of Voevodsky (particularly his category $DG$). It would be nice if somebody could explain that to me.
I addition I am very keen on seeing how these ideas can be used in an explicit example. If I should explain someone why motivic cohomology is a good thing, I would certainly mention the proof of the Milnor conjecture. But can one see the use of derived ideas in a more explicit and down-to-earth example?
 A: They are very different.  Both involve a mixing of algebraic geometry and homotopy theory, but not at all in the same way: DAG is when you use more homotopyish rings for your basic affines, whereas in Voevodsky/Morel's work you're thinking of a variety as a kind of space and trying to capture the homotopy type of that space in a universal way.  Even more loosely, in DAG you're injecting the heroin of homotopy theory right into the foundations of algebraic geometry, keeping the same form; but in motivic homotopy theory you're trying to push varieties into that decadent realm, loosen them up a little, and you fundamentally change their form.
A: There is a very general nice pattern here:
Let $C$ be a category of test spaces on which we want to model more general spaces. Then 


*

*a "very general space" modeled on $C$ is an object in the gros ∞-topos $Sh_{(\infty,1)}(C)$ of ∞-stacks on $C$.


Morel-Voevodsky take C to be the Nisnevich site. Then $Sh_{(\infty,1)}(C)$ is the the ∞-topos whose intrinsic cohomology is motivic cohomology.
Here C happens to be just an ordinary category. More generally, we could take C to be an ∞-category itself. In that case $Sh_{(\infty,1)}(C)$ could be called the ∞-topos of derived stacks. What Toen Vezzosi do in HAG I and II is to provide a model-cateory theoretic presentation of this. That's why the articles look like they are hard to read: this is a component based way to describe an abstract elegant concept.
Now, the objects in $Sh_{(\infty,1)}(C)$ are "very general" spaces modeled on $C$. There is a chain of ∞-subcategories of more "tame" spaces inside, though: 


*

*first there are those ∞-stacks on C which are represented by an ∞-stack with a C-valued structure sheaf. This are the structured ∞-topos, that generalize the notion of ringed spaces.

*and then among these are those that are locally equivalent to objects in C. These are the generalized schemes or "derived scheme" if C is suitably ∞-categorical.
The pattern here is reviewed at notions of space.
In principle one could consider such "derived schemes" also in the Morel-Voevodsky gros ∞-topos of ∞-stacks on the Nisnevich site, it just seems that so far nobody looked into this. But there are all kinds of examples of test space categories C that people still have to push through this general nonsense. For instance we can take C to be simply the category of smooth manifolds. Then the above generalities spit out the notion of a derived smooth manifold.
The punchline being: all these things unify in one grand picture. It is not Toen/Vezzosi/Lurie derived geometry on one hand and Morel/Voevodsky cohomology on the other. Instead all this is parts of one big picture.
And also, if I may say this here, this big picture really is hardly restricted to algebraic geometry. Lurie's notion of space is much, much more general. It describes GEOMETRY. Of whatever sort.
A: DAG is Jacob Lurie's thesis.  It also stands for the whole subject called "Derived Algebraic Geometry".  It is the use of stable infinity-categories to construct generalized geometric spaces.  So in particular, we have higher algebraic stacks, higher descent, higher toposes, higher moduli spaces, etc.
Voevodsky's work that you mentioned "lives in" this world of derived algebraic geometry.  In particular I'm thinking of his usage of simplicial presheaves.  
An example that I can think of... hmm.. 
This isn't the best one, but it turns out that triangulated categories are just categories that "look like" homotopy categories of stable infinity categories.  In this setting, it turns out that the very unnatural octahedral axiom becomes a natural consequence of the higher categorical data.
The derived category of the category of chain complexes on an abelian category is actually the same as the homotopy category of the stable infinity category of chain complexes.  
This kind of explains what people mean when they say that homology and cohomology naturally live in higher categories.  
A: I can't tell you much about the relation to Voevodsky's work, but I can give you a quick summary of Derived Algebraic Geometry. 
In DAG you enlarge the category of geometric objects that you can study. It is easiest done by using the functorial approach. So a scheme is just a functor from commutative rings to sets. 
Now we'll first enlarge this category in the stacky direction. This means that we change the target of our functor. Instead of studying only set-valued functors we'll allow groupoid-valued functors. This leads to the stacks you are used to. But for some complicated moduli problems groupoids can't encode enough information. So we'll make the target category even bigger and allow for simplicial set valued functors. So now you have to study the category of functors from commutative rings to simplicial sets, also called the category of simplicial presheaves. In this category you have to impose the right descent and atlas conditions to find the objects that have the right to be called geometric. These guys will be called higher stacks. 
I think that what we have done so far is very similar to Voevodsky's construction. But I am absolutely not an expert on this. Probably one of the experts will show up here soon and explain that. 
So far we haven't derived anything. That starts when you also enlarge the domain category. The natural "derived category" for commutative rings is simplicial commuative rings. That's because the category of commutative rings is not abelian, and then simplicial objects are a good replacement for chain complexes. Derived algebraic geometry then is the study of functors from simplicial commutative rings to simplicial sets, or simplicial presheaves on simplicial commutative rings. Again you have to find the functors inside this category with the right descent and atlas conditions, and that's a lot of work. These guys are then called derived schemes, derived stacks and derived higher stacks. 
The geometric intuition for these derived schemes and stacks are that they are ordinary schemes plus a fuzzy cloud of nilpotents on steroids around them. They can encode much more information in their structure sheaf than ordinary schemes. A good example is the intersection of two subschemes in an ambient scheme. You can then construct a "derived intersection". This derived intersection intrinsically in its structure sheaf has encoded that it is an intersection, something you could never accomplish with normal nilpotents. 
The difference between the Toen-Vezzosi approach and the approach of Lurie is that TV use model categories where Lurie uses infinity-1 categories. Which approach you actually use is a matter of taste. I think the analogy is to either work coordinate free or with coordinates. 
A final comment: If you look into the papers on Luries website or into Toen-Vezzosis Homotopical Algebraic Geomtery II book, you'll find that they work in much greater generality. They do the whole program not on the category of simplicial commutative ring, but for quite general model categories. If you really are only interested in DAG, there are Toen's course notes on his homepage or Luries original thesis. 
A: As far as I understand (which is not far with the Voevodsky stuff) there are three differences between the two directions:


*

*Motivic homotopy theory concerns sheaves of spaces in the Nisnevich topology, which is coarser than the etale or flat. This means we can have more objects here that are not allowed in the DAG/stacky world. For example algebraic K-theory satisfies Nisnevich but not etale descent, so doesn't live in the stacky world. It would be great if someone could give an intuitive feel for how much more general Nisnevich sheaves can be than etale ones - so as to get a sense of how fundamental this difference is.

*As mentioned by several people, in DAG we can extend from stacks over rings to stacks over derived rings of one kind or another. This is important for applications but not a fundamental difference with motivic homotopy theory, since one can certainly imagine combining the two as Urs says.

*The most fundamental difference IMHO (which seems missing from all but Charles' response) is inverting the affine line - passing to $\mathbb A^1$ homotopy theory (though presumably this is implicit in Dustin's metaphors). This fundamentally changes the nature of the geometric objects in motivic homotopy theory, making them far more flexible and "homotopic" than those of DAG (compared to this presumably the extra flexibility of working in Nisnevich topology is just a minor step?)
Of course one CAN probably consider such operations in DAG, as Charles suggests, but in terms of the current flavors and applications of the two areas this seems to me a huge difference. It is in this sense that motivic homotopy theory is really doing homotopy theory for schemes -- in the theory of DAG and higher stacks we keep the scheme direction alone (or allow it to be infinitesimally thickened in a derived direction) and only homotopify the values of functors of points, while $\mathbb A^1$ homotopy adds a completely different kind of topology into the game.
A: I'll add to the other answers with a speculative comment:
Presumably, it should be possible to "do Morel-Voevodksy in DAG".  Thus, start with some convenient category of derived schemes $C$, define a notion of Nisnevich sheaf on $C$, pick an object $I$ in $C$ (the affine line?), and localize by forcing $I\to *$ to be an equivalence.
I have no idea if this is an interesting thing to do.
A: In answer to the last part of your question about explaining why motivic homotopy theory is a good thing, check out my recent question about applications of algebraic K-theory to number theory. Both the question and answer make reference to motivic homotopy and how it helps with the study of K-groups. In particular, you could use Vandiver´s Conjecture and the other applications of K-theory as good reasons to care about motivic homotopy theory.
