Tightening Zhang's bound Inspired by a blogpost by Scott Morrison and ongoing discussion there I decided to create this community wiki to track progress on the original bound of Yitan Zhang.
The original bound was $70,000,000$. The accepted answer should contain latest known improvement.

As of 4.6. 2013 there is a polymath project devoted to improving this bound. The proposal can be found at http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/
Links to various references: http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes
A good place to start is to read notes by Terence Tao and his blog post on the topic.
 A: Original approach
A set of integers $H$ is called admissible if it avoids at least one residue class modulo $p$ for each prime $p$. In other words
$$
\forall p \in \mathcal{P} :\text{cardinality of} \, \lbrace x \bmod p \, | \, x \in H \rbrace \leq p-1.
$$
Let $Q(k_0)$ denote the assertion that for any admissible set $H$ of cardinality $k_0$ there are infinitely many translates $n+H$ that contain at least two primes. The bound on the gap is then $\mathrm{diam}\, H$.
Zhang deduces his bound from the following result:
T1: $Q(3,500,000)$ is true
In Zhang's paper the length $k_0$ is determined by the following inequality (1) that has to hold for some natural number $l_0$
$$
(1+4\varpi) (1-\kappa_2) > \left(1 + \frac{1}{2l_0+1}\right) \left(1 + \frac{2l_0+1}{k_0}\right) (1+\kappa_1),
$$
where
$$
\kappa_1 = \delta_1 \left( 1 + \delta_2^2 + k_0 \log\Bigl(1+\frac{1}{4\varpi} \Bigr) \right) \binom{k_0+2l_0}{k_0}
$$
$$
\kappa_2 = \delta_1 (1+4\varpi) \left(1 +\delta_2^2 + k_0 \log\Bigl(1+\frac{1}{4\varpi} \Bigr) \right) \binom{k_0+2l_0+1}{k_0-1}
$$
$$
\varpi = 1/1168
$$
and
$$
\delta_1 = (1+1/4\varpi)^{-k_0}
$$
$$
\delta_2 = \sum_{j=0}^{1/4\varpi} \frac{k_0\log(1+\frac{1}{4\varpi}))^j}{j!}.
$$
The admissible set that Zhang uses is $H = \{ p_{k_0+1}, \ldots, p_{2k_0}\}.$
Current record
Terence Tao & Scott Morrison: 4,802,222 
Terence Tao established another inequality on $k_0$ that manages to remove most of inefficiency of Zhang estimate.
$$
1+4\varpi > \left(1 + \frac{1}{2l_0+1}\right) \left(1 + \frac{2l_0+1}{k_0}\right) (1+\kappa)
$$
where
$$
\kappa := \sum_{1 \leq n < 2 + \frac{1}{2\varpi}} \Bigl(1 - \frac{2n \varpi}{1 + 4\varpi}\Bigr)^{k_0/2 + l_0} \prod_{j=1}^{n} \left(1 + 3k_0 \log\Bigl(1+\frac{1}{j}\Bigr)\right).
$$
Moreover $l_0$ is allowed to be real number. Scott Morrison then found that for $l_0 = 291.7$ one gets $k_0 = 341,640$ which is the best possible $k_0$ for given $\varpi = 1/1168$.
Paper by Richards suggest to take as admissible set $H_m = \{ \pm 1, \pm p_{m+1}, \ldots, \pm p_{m+k_0/2+1} \}$ for $m$ large enough. This leads to bound 
$$
2p_{m+\lceil k_0/2 \rceil + 1} \quad \text{for }  k_0  \text{ even}
$$
and 
$$
p_{m+\lfloor{k_0/2}\rfloor-1} + p_{m+\lfloor{(k_0+1)/2}\rfloor-1} \text{ for } k_0 \text{ odd.}
$$
For given $k_0=341,640$ program written by Scott Morrison found that $m=5553$ gives the smallest bound of $4,802,222$.
