Is it feasible to detect light emitted from planet earth 2,000 years ago by reflection from an object situated 1000 light years away? The possibility though remote, is intriguing as we may be able in the future to actually "see" our own planet's history.
Though sounding science fiction, if we are able to detect bodies in space that are able of reflecting light emitted from our planet earth, using amplification systems and filters,this may - if possible, give us a tool of utmost importance.
 A: Not in any practical sense. First of all, the intensity will be very low: inverse-r-squared going out to the reflector, and inverse-r-squared coming back. As @WillCross pointed out, it will be at very low levels. Secondly, the resolution will be very poor, as there's nothing to focus it and the subtended angle is very very small (in both directions).
Consider that we already do have such a reflector, about 1.5 light seconds away: the dark side of the Moon, illuminated by Earthshine. It somewhat lights up the lunar surface, and variations in the total illumination can be detected that correlate to cloud cover (increased reflection) on Earth. Even being so close, there's no way to see any sort of useful images as you envision. Maybe you could detect a massive nuclear war by reflections from the Moon or nearby planets, but at light year distances, it would be hard to sort out from the noise.
A: The average albedo of the Earth is about 0.3 (i.e. it reflects 30 percent of the light incident upon it).
The amount of incident radiation from the Sun at any moment is the solar constant ($F \sim 1.3 \times 10^3$ Wm$^{-2}$) integrated over a hemisphere. Thus the total reflected light from the Earth is $L=5\times 10^{16}$ W.
Let's assume this has the same spectrum as sunlight and let's assume that this light gets reflected from something which is positioned optimally - i.e. it sees the full illuminated hemisphere. In that case, roughly speaking, the incident flux on any reflecting body will be $L/2\pi d^2$ (because it is scattered roughly into a hemisphere of the sky).
Now we have to explore two divergent scenarios.


*

*There just happens to be a large object a 1000 light years away that is highly reflective. 


Let's be generous and say it is a perfect reflector, but we can't assume specular reflection. Instead let's assume the reflected light is also scattered isotropically into a $2\pi$ solid angle. Thus the radiation we get back will be
$$ f = \frac{L}{2\pi d^2} \frac{\pi r^2}{2\pi d^2} = \frac{L r^2}{4\pi d^4},$$
where $r$ is the radius of the thing doing the reflecting.
To turn a flux into an astronomical magnitude we note that the Sun has a visual magnitude of $-26.74$. The apparent magnitude of the reflected light will be given by
$$ m  = 2.5\log_{10} \left(\frac{F}{f}\right) -26.74 = 2.5 \log_{10} \left(\frac{4F \pi d^4}{L r^2}\right) -26.74 $$
So let's put in some numbers. Assume $r=R_{\odot}$ (i.e. a reflector as big as the Sun) and let $d$ be 1000 light years. From this I calculate $m=85$.
To put this in context, the Hubble space telescope ultra deep field has a magnitude limit of around $m=30$ (http://arxiv.org/abs/1305.1931 ) and each 5 magnitudes on top of that corresponds to a factor of 100 decrease in brightness. So $m=86$ is 22 orders of magnitude fainter than detectable by HST. What's worse, the reflector also scatters all the light from the rest of the universe, so picking out the signal from the earth will be utterly futile.


*A big, flat mirror 1000 light years away.


How did it get there? Let's leave that aside. In this case we would just be looking at an image of the Earth as if it were 2000 light years away (assuming everything gets reflected). The flux received back at Earth in this case:
$$ f = \frac{L}{2\pi [2d]^2} $$
with $d=1000$ light years, which will result in an apparent magnitude at the earth of
$m=37$.
OK, this is more promising, but still 7 magnitudes below detection with the HST and perhaps 5 magnitudes fainter than might be detected with the James Webb Space Telescope if and when it does an ultra-deep field. It is unclear whether the sky will be actually full of optical sources at this level of faintness and so even higher spatial resolution than HST/JWST  might be required to pick it out even if we had the sensitivity.


*Just send a telescope to 1000 light years, observe the Earth, analyse the data and send the signal back to Earth.


Of course this doesn't help you see into the past because we would have to send the telescope there. But it could help those in the future see into their past.
Assuming this is technically feasible, the Earth will have a maximum brightness corresponding to $m \sim 35$ so something a lot better than JWST would be required and that ignores the problem of the brightness contrast with the Sun, which would be separated by only 0.03 arcseconds from the Earth at that distance.
Note also that these calculations are merely to detect the light. To extract anything meaningful would mean collecting a spectrum at the very least!
A: There are several reflectors that were placed on the moon during the Apollo mission and the number of photons detected (after being reflected) is very small compared to the number of photons the laser emits. See the section "Details"
A: Rather late, but as LDC3 resurrected this question from 2013....
This has already happened. Radio waves reflected from what is presumed to be an asteroid swarm about 25 LY away have returned to earth and been identified as early TV broadcasts. Quite appropriately, the first identified signals were episodes of Dr. Who.
However, radio and visible light are rather different things. If you are expecting to watch Julius Caesar walk up the steps of the Portico, it won't happen. We can barely do that from low orbit.
