Q:

Mars and back, staying only a few days

This article states that

once you arrive at Mars you'll only have a couple of weeks before you'd have to begin a fresh Hohmann transfer back to Earth, otherwise you'll have to hang out on Mars for about 18 months before another launch window comes around, to return to Earth! That's why human missions to Mars are either short term (ten days or so on the planet) or long term (two years on the planet).

This sounds like you can go to Mars and back to Earth within a single Hohmann transfer. (I presume leaving the transfer orbit for about ten days would not significantly change the overall trajectory.)
However, the semi-major axis of a Hohmann transfer between the orbits of Earth and Mars, given by $$\frac{r_\mathrm{Earth} + r_\mathrm{Mars}}{2}\text{,}$$
is about 1.26 AU (assuming Earth and Mars have circular orbits around the sun, with $r_\mathrm{Earth} = 1\,\mathrm{AU}$ and $r_\mathrm{Mars} = 1.52\,\mathrm{AU}$), for which Kepler’s 3rd law gives an orbital period of 1.42 years. Hence, when an object following this Hohmann orbit arrives, after departing from Earth, in Earth’s orbit again for the next time, almost one and a half year will have passed and Earth will be somewhere on the other side of the sun: The transfer orbit would not return to Earth (at least not within one orbital period).
Neither Earth’s nor Mars’s orbit is exactly circular, and there is a difference in inclination, so the transfer orbit will not be exactly a Hohmann orbit, but the relative inclination and orbital eccentricities of Earth and Mars are low, so I do not expect this to have a significant impact.
I am also aware that a transfer orbit between two planets does not necessarily have to be a (near-)Hohmann orbit. However, as I understand it, there is a considerable difference in terms of energy (and thus fuel) requirements, making any transfer other than (near-)Hohmann more or less infeasible (in most situations, and specifically for Earth–Mars or back).
Given that a single Hohmann transfer orbit from Earth to Mars does not return to Earth, how would a Mars mission with staying only about ten days on Mars (as described by the article) be accomplished? Can it (reasonably) work at all, or is that article wrong? It also claims that

the Hohmann Transfer Orbit […] takes a good six months to go from Earth to Mars

which does not match the calculation above (the half of 1.42 years is 8.5 months), so I suspect that it either refers to some other kind of transfer orbit or just got the numbers wrong.

A:

There is a popular idea that has been put forward that one can return to Earth via Venus, and to achieve that one has to launch within a few weeks of landing on Mars. I suspect that the author is confusing that. You can take a look at http://clowder.net/hop/railroad/sched.html for when the times are, and see that the times pretty closely overlap.
Theoretically one could take a very fast journey there, that is slightly non-Hohmann, and do the same in return. This could allow for a brief period of stay. 
Note that a perfect Hohmann orbit between Earth and Mars does not really exist. That would require no inclination difference, and no eccentricity. But there are a number of good orbits that approximate Hohmann times, but the time to and from won't be as clean as the perfect solution proposed.

