Update: (Oct. 2018) For the first time, all ICM 2018 lectures (plenary, invited and special) as well as panels and special events are presented (by good-quality videos) on the ICM 2018 You tube channel.
Update:(Dec 2017) The ICM launched a new website. All previous ICM proceedings are available here. However, I cannot find the old page with access to individual papers and search options.
Just recently the International Mathematical Union (IMU) put online all the previous proceedings on the ICM's! Here: (update:broken). This webpage is based on joint work by R. Keith Dennis (Ithaca) and Ulf Rehmann (Bielefeld). So you can read for free all the articles (including Hilbert's famous problem paper; Martin Grötschel demonstrated it at the opening ceremony of ICM 2010). I don't know when the proceedings of ICM2010 will be added. UPDATE: the 2010 articles have now been added! Update The ICM2014 papers are available here (you can download each of 4 volumes; the IMU site does not contain these papers yet.)
As for the talks themselves, there is a page with all the ICM 2010 plenary talks here; links for videos from earlier ICM's (plenary talks and other events) can be found here.
Update (August, 6 2014) Many (46 for now) of the ICM 2014 proceedings contributions are already available on arXiv, via this search. (I got it from Peter Woit's blog.) Videos of lectures can be found here.
Update (May 2014): Starting 1992 there is also every four years the European Congress of Mathematics (ECM) that the European Mathematical Society (EMS) is running.
The proceedings of the first three ECMs are now freely open. These volumes are available here.
(Digitising the proceedings of the first three ECMs, published by Birkhäuser was a task carried out by the EMS Electronic Publishing Committee.)
Starting with the 4ECM (2004), the Proceedings are published by the EMS Publishing House. The EMS decided to make them freely available online too. I expect that this will happen soon and I will keep you posted. Further update (June 2014) The ECM Proceedings are now available here! I was told that in a few months, the EMS will put also the 6ECM volume.
I was attending ICM 2002 in Beijing, in my first postdoc year, which was wonderfully organized. I expected that as a general mathematics meeting, the talks should be very accessible, but even plenary talks (with few exceptions, like the talks by Mumford and Hopkins) were accessible (for nonexperts) only about first 15 minutes or so, depending on your attention stamina in the midst of flood of data and the background. My background is reasonably wide, and I am used to find myself well at conferences in a number of area, but most of the talks at ICM were too detailed and fast. Some plenary talks had of the order of 45 transparencies. During the breaks people run in other lecture rooms, and in the mass of people people get lost around. I would recommend knowing and contacting in advance some people you like to hang out, improvising is more difficult at such big conferences than in small ones, which are mathematically good for me. It also depends on how far is the hotel from the site.
India is a wonderful country, which I visited in 2007; the summer is a bit too hot though Hyderabad is slightly in the hilly and drier area than big cities like Madras and Kolkatta; in India hygiene is often a problem (and much worse than in China), especially with vegetables and water (UV-filtered water which they use is OK, you do not need bottled water though). I brought a bottle of home made brandy and was taking a sip few times a day for precaution and never had a stomach problem, unlike most westerners.
I have math/physics friends in India, and if Croatian science funding were not in crisis (and still declining) I would go. I do not know your background, but if you come unprepared, without good acquaintances and so on, be aware of hectic atmosphere of big congresses and the needs of the climate, hygienic and other adaptation in subtropics.
Best Answer
The logo has a piece of the complex upper half plane divided into fundamental domains for the action of $SL_2(\mathbb{Z})$ by Möbius transformations (which are hyperbolic isometries - see the appropriate entry in McMullen's gallery or Wikipedia). There are no Ford circles in sight, but you may have been confused by the semicircular regions on the bottom. Those regions are unions of fundamental domains, and are cut out by geodesics in the $SL_2(\mathbb{Z})$ orbit of $1/2 \leftrightarrow \infty$ (which forms part of the boundary of a few fundamental domains). Ford circles come from the $SL_2(\mathbb{Z})$ orbit of the horocycle $\operatorname{Im}(\tau) = 1$, and horocycles have constant nonzero geodesic curvature (imagine driving with your steering wheel turned a bit to the right, but never returning to where you started).
The generating function for $\tau$ is the 24th power of Dedekind's $\eta$ function (often written as the discriminant form $\Delta$), and it is a function on the upper half plane that is invariant under the weight 12 action of $SL_2(\mathbb{Z})$. Up to normalization, it is the unique lowest weight nonzero level 1 cusp form, and this makes it automatically a Hecke eigenform. Both Mordell's proof of the earlier Ramanujan conjectures and Deligne's proof of the growth conjecture use this fact in an essential way. I should note that Deligne proved the Ramanujan conjecture as a corollary of the Weil conjectures, not "in the process".
The connection between hyperbolic geometry and the Ramanujan conjecture is not particularly strong, as far as I know (but I would be happy to be shown the errors of my ways).