[Update: the video has been removed the by the owner. I'm not sure why, so I dropped him an email about it. Stay tuned!]
This is pretty cool: back in July 2007, an amateur astronomer made a video of the International Space Station as it passed directly in front of the Sun:
There's a lot to note here:
1) Most obvious is the speed of the ISS. It orbits the Earth a mere 350 or so kilometers (220 miles) up; I like to say that if you live in DC and see it pass overhead, it's about the same distance from you as New York City. So it's actually pretty close to the Earth's surface, and screaming around at 8 km/sec (5 miles/sec). That's a good clip! From the point of view of someone watching from the ground, it only takes a couple of minutes for the station to go clear across the sky, horizon to horizon.
Also, the Sun is pretty small in the sky; you can easily cover it with your outstretched thumb. So the great speed of the ISS coupled with the small apparent size of the Sun means the entire pass will take less than a second! 2) The astronomer -- Len Marek -- wasn't using a huge telescope to get this shot. He has a Coronado Maxscope 60, which has a 6 cm (2.4 inch) lens. It's a very nice telescope, equipped with a filter that only lets a very narrow range of red light through, and is a bit pricey, but it's not some huge honking 'scope. It doesn't take something very big to see details on the Sun.
3) The encounter of ISS and Sun was predictable! As he says in the video notes, he used some software (Cal-Sky) to discover the ISS would pass in front of the Sun at a given time, and he was ready to take the video. There are any number of packages that allow you to do this.
I'll add you don't even need a telescope to see the ISS -- go to Heavens Above, enter your latitude and longitude as accurately as you can (Google maps might help here) and you're all set! It will tell you about good ISS passes, as well as other satellites that might be visible from your location. The ISS can be brighter than Venus on a good pass, making it the third brightest object in thenight sky. And if you have good binoculars, you might even be able to see a bit of detail in the structure, like the fact that it's elongated.
Astronomy is one of those cool fields that can be enjoyed by everyone from rank beginner to seasoned professional. And you don't need a lot of fancy equipment. Just your eyes and some preparation is sometimes all you need.
Tip o' the Hα filter to reddit.