Started in 1998 and continuously occupied since November 2000, the ISS is currently the largest space station ever constructed, at 120yds by 80yds - larger than a soccer pitch - and weighing around 415 tons. The picture here shows it in late March 2009. For further facts about the ISS, click here.
The ISS's large flat solar-cell arrays are excellent reflectors of sunlight, with the result that it can appear as a very bright object in the sky. Even so, it will only be visible with the naked eye close to sunrise or sunset: during the night there will be no sunlight around to illuminate it [but see below!] and during the day there will be too much to allow it to be seen. In addition, it will not be visible every day because [at the time of writing this page - Sept 2010] it takes just over 91mins to orbit the Earth and so after 24hrs, when the Earth has revolved exactly once, the ISS will have made just less than 16 orbits. In the 22mins or so the ISS needs to complete the last orbit the Earth will rotate just over 51/2deg further, which will eventually take the observer out of the visual range of the orbit track. This will take a while though, as the "circle of visibility" has a radius of well over 1000mls - this means you can see the ISS in the UK when it is actually passing south of the Alps! However, while this shift may take the ISS out of range for an evening pass, for example, it may well be taking it back into range for a morning pass. There will thus be distinct "seasons" of visibility: a run of days when it can be seen in the evening and then a sequence in the morning, separated by a period of no visibility when the orbit track is just too far away. During a "season", however, it may be possible to see it more than once on a given night - if the first pass is early in the evening for example, 91mins later there may still be enough sunlight to illuminate it a second time. Conversely, the station may pass into the Earth's shadow before it goes out of view below the horizon. As with the same phenomenon when it involves the Earth's natural satellite the Moon, this is called "going into eclipse".
More than two consecutive passes may be visible in the summer-time as then the sun isn't very far below the horizon and so will be more able to illuminate the station. This means of course that the evening and morning sequences of passes tend to merge, thus making the concept of a visibility season less meaningful in the summer. The maximum number of consecutive passes is five because firstly there isn't much more than 6hrs (91mins times 4) of darkness in summer anyway and secondly the geometry of the circle of visibility dictates that if the first pass just clips the circle to the south-east the fifth will just clip it to the south-west i.e. five consecutive passes (and no more) may all just have some portion of the track visible. This fact also allows us to calculate an estimate for the maximum length of a visibility "season". If it takes 4 orbit periods (365.6mins) for the track to cross the circle of visibility then, given that in one day the ISS falls 22.4min behind the observer, it will be visible for 365.6/22.4 days i.e. 16 or 17 days. This is actually a pretty good estimate for a "typical" season but because the geometry of a pass is a bit more complicated than indicated above (it's in 3 dimensions, not just 2, for example), and at the very end of a season an earlier orbit than the one the calculations are based on can sneak into view instead, marginal cases can sometimes be just visible (technically!), thus extending the season. The greatest duration I found in winter 2007/8 was 19 days, though the first and the last three passes (over two days) fell into the marginal (or exceedingly marginal!!) category with visibilities of 35sec, 4sec, 26 sec & 6sec.
The orbit of the ISS is inclined by 51.6deg to the equator and so for locations with latitude greater than this (i.e. most of the UK) it can never pass directly overhead. This means that its track across the sky only really varies in maximum altitude as an observer must always look towards the South to see it. Given that it orbits in the same direction as the Earth rotates, it therefore tends to rise in the West, attain maximum height in the South and then set in the East (although its actual visibility may be affected by eclipse). Passes that are only just within the circle of visibility will differ from this basic plan somewhat but as they will also be quite low down (and thus more difficult to observe) this isn't really important.
The brightness of the ISS depends on the angle at which it is "flying" relative to the observer and to the sun (as this determines how reflective it is) and on the distance from the observer. The angles vary quite a lot because its attitude is actively controlled by on-board systems to, amongst other things, maximise voltage generation by ensuring the solar cells are pointing towards the sun. Also, because each orbit passes through the circle of visibility differently, even if the ISS itself didn't move it would be presented to the observer (and the sun) at a different angle each time. The straight-line distance at the point of closest approach varies from about 250 to over 800mls but when just rising or setting it can be as much as 1350mls away. All this means that its maximum brightness on a pass currently ranges from about magnitude -0.2 to -3.5: a variation by a factor of over twenty! Passes which are easily visible (i.e. quite high in the sky) are very much towards the top end of the range, however.
The other thing that is affected by distance is the speed at which the ISS appears to move. When it is low down (and thus far away) it moves really slowly but as it gets nearer and nearer its speed across the sky increases considerably so that at maximum elevation it can be quite difficult to track with a camera or through binoculars.
