What is the name of the distribution with a probability density like $1/(1+\exp(x))$? Please forgive my ignorance, what is the name of the distribution with a probability density like this?
$$p(x) \propto \frac{1}{1 + e^x},\quad x > 0\,,$$
or more generally
$$p(x) \propto \frac{1}{1 + \alpha e^{\beta x}},\quad x > 0\,,$$
or
$$p(x) = \eta \frac{1}{1 + \alpha e^{\beta x}},\quad x > 0\,,$$
where $\eta$ is a normalising constant.
 A: The normalizing constant for the first should be $\frac{1}{\ln(2)}$ (not that it really matters for the present question).
I'm not aware of either having a name. The first (without the $\log(2)$ normalizing constant) is the survivor function for a truncated logistic distribution, but I haven't seen it used for a density function (though I expect that it has probably been named several times ... that's often the case with simple functional forms that are not in very wide use, where people "reinvent" such things without encountering previous ideas, which are often in different application areas*).   

If you were to try to name it, then because of the logistic-type functional form you'd probably want to squeeze the word "logistic" in there somewhere, but the difficulty would be in choosing a name that would distinguish it sufficiently from the logistic density.
* and jwimberly's answer offers one such application area. The name "Fermi-Dirac distribution" seems a perfectly reasonable choice if you don't have a name in the application area you're working in.
A: This is identical to a common distribution in physics called the Fermi-Dirac distribution, which describes a situation called Fermi-Dirac statistics. In a certain setting in physics, the average number of particles with an energy $\epsilon$ is
$$
\bar{n}_\epsilon = \frac{1}{e^{(\epsilon -\mu)/kT}+1}
$$
where $\mu$, $k$, and $T$ are physical parameters that probably aren't so important to you (the chemical potential, Boltzmann's constant, and the temperature). Its trivial to reinterpret this as probability density function for the energy of a particle.
A: A density that integrates to unity over $[0,\infty]$ would be
$$f_X(x) = \frac {\theta}{\ln 2}\frac{1}{1+e^{\theta x}},\;\;\; \theta >0$$
Raw moments are given by 
$$E(X^k) = \frac {(1-2^{-k})}{\ln 2} \frac {1}{\theta^{k}}\cdot \Gamma(k+1) \cdot \zeta(k+1)$$
where $\Gamma()$ is the Gamma function and $\zeta()$ is the Riemann zeta function. So
$$E(X) = \frac {\pi^2}{12\cdot \ln 2}\theta^{-1} \approx 1.1866 \cdot \theta^{-1}$$
$$E(X^2) \approx   \frac {7.212}{4\cdot \ln 2}\theta^{-2} \approx 2.601 \cdot \theta^{-2} $$
leading to 
$$\text{Var}(X) \approx 1.193 \cdot \theta^{-2}$$
Numerical calculations verify these.
