Why is the eclipsed conformer more stable than the bisected conformer in 1-butene?
In alkanes staggered conformers (all anti) are local minima, while eclipsed conformers represent transition states. The first is "stable", while the latter is not. The rotational barrier has been measured and calculated often, it is around 12&nbsp;kJ/mol.[1]

In alkenes the eclipsed conformers are local minima, while the staggered, or more accurately bisected, conformers are transition states. The rotational barrier for propene is only about 8&nbsp;kJ/mol.[2]

I have performed a full, relaxed rotational scan of 1-butene around the central carbon-carbon bond axis (blue). The minima and maxima of this scan have been reoptimised and thermal corrections at 298.15&nbsp;K and 1&nbsp;atm have been added (orange).



The chosen level of theory, DF-BP86+D3(BJ)/def2-SVP, is not the most accurate, but it will give you a general idea. You can see the bisected conformations are maxima along the rotational coordinate and the eclipsed ones are minima.
There is dispersion correction included, which makes the E1 conformer slightly more stable than the E2 conformer.
Including the thermal corrections, the energy difference between these two structures is negligible.
$$\begin{array}{crrr}
\text{Conformer} &amp; 
\angle_\mathrm{D} &amp; 
\Delta E~/\mathrm{kJ\,mol^{-1}} &amp;
\Delta G~/\mathrm{kJ\,mol^{-1}}\\\hline
\mathrm{E_1}     &amp;   0.0             &amp;     0.0  &amp;   0.0   \\
\mathrm{B_2}     &amp;  60.0             &amp;     9.9  &amp;  10.1   \\
\mathrm{B_1}     &amp; 120.0             &amp;     1.2  &amp;  -0.2   \\
\mathrm{E_1}     &amp; 180.0             &amp;     9.7  &amp;   9.4   \\\hline  
\end{array}$$

The eclipsed forms are stabilised by two &pi;-&sigma;/&pi;-&sigma; hyperconjugation interactions. Here is the visualisation for E1 in the natural bond orbital picture. Occupied orbitals are depicted blue/orange, virtual orbitals are depicted red/yellow. The phases are assigned randomly.
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In the B2 conformer the overlap between those orbitals is not ideal. The nodal planes in the anti-bonding orbitals are in the way.
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The representations for E2 are analogue to E1, and B1 is similar to B2. Because of time reasons I have not been able to create the pictures yet.

Taking the maximum difference in the energy we can approximate the rate of the reaction with transition state theory (assuming $\kappa=1$):
$$k = \kappa\frac{k_\mathrm{B}T}{h}e^{\frac{- \Delta G^{\ddagger }}{RT}}
\approx 1\times10^{11}~\mathrm{\frac{1}{s}}$$
This is ultrafast and at room temperature we consider it a free rotation. With spectroscopic methods we will not see a difference. We would probably have to cool the system down to below 1&nbsp;K to actually see two (E1, E2) conformations.

Steric effects are often electronic effects in disguise, they offer tempting explanations, but are often rooted in a more complex electronic explanation. In this example the possibility of hyperconjugation explains the slightly higher energy the bisected forms compared to the staggered forms.
Hyperconjugation can come in different forms, the most common explanation is the overlap of an empty anti-bonding orbital with a filled (non-)bonding orbital. In this scenario it does not matter which elements are involved in these interactions.
There is even negative hyperconjugation, but this is a little beyond the scope of this post - if there are more questions about the topic, you are always welcome to ask them here.



Footnotes


For a quite recent molecular orbital study see: Ramiro F. Quijano-Quiñones,  Mariana Quesadas-Rojas,  Gabriel Cuevas and  Gonzalo J. Mena-Rejón, Molecules 2012, 17(4), 4661-4671. (open access)
For an educational approach see: Henry Eyring, D. M. Grant and H. Hecht
J. Chem. Educ. 1962, 39 (9), 466.
I did not have the time to search thoroughly, but the following publication contains the value. Leland Cullen Allen, Eugene Scarzafava, J. Am. Chem. Soc. 1971, 93 (2), 311–314.