SpaceWell, the solar sail launch failed; the poor old converted ICBM didn't make it. Probably. Maybe. Reports seem confused. And if it didn't, well, that's the third failure. Who knows, maybe they'll try again...
So why are we still using chemical rockets? It's funny how little basic Earth-to-orbit delivery devices have changed since the launch of Sputnik almost 50 years ago. Werner Van Braun, who designed the V2 and subsequently worked for the Americans, designed an orbital launch craft during WW2. Germany even had vague plans to BUILD it.
And indeed, the first spacecraft were launched on what were basically glorified dual-stage V2s. And then the next spacecraft, and the next... The Soviet Union developed two vaguely interesting launch vehicles early on; the Soyuz and the Proton. The Proton was a heavier-lift rocket, and used(uses) a highly-toxic fuel which doesn't need ignition. The US got started a bit slower, but did more or less the same thing.
Back then, elaborate launches-from-air and spaceplanes and so on were looked at, but the giant rocket design was already being developed to carry nuclear missiles, and both countries had gained great knowledge of it from V2s and scientists taken from Germany. It was the obvious way to go.
When the moon race started, both countries looked at more innovative solutions, but in the end went back to the chemical rockets; after all, they were known territory. The US built first the Saturn I, then the huge Saturn V, 100m tall and with ability to lift 110,000kg into a low Earth orbit. 15 were built, and 13 used, one carrying the Skylab space station. Even larger rockets were looked at; a larger Saturn V and various huge "Nova" designs, but the age of large rockets was wrapped up with unseemly haste after Skylab, leaving only the much smaller Titan (18,000kg to LEO) The Russians tried to build something similar, but their overcomplicated 30-jet first stage tended to explode.
In response to the American space shuttle, Russia went back on the unreasonably big rockets thing. Energia came in three configurations; a "small" 2-booster version (used once for a failed military platform launch), a 4-booster 4-core (it's like the semiconductor industry, really, isn't it?) version, used to launch the Russian shuttle, Buran, and a huge 8-booster version, with the 2 booster version as an upper stage. This was big enough to launch 175,000kg to LEO, and big enough for a manned mission to the moon, establishment of a manned base on the moon, and even a one-rocket manned Mars mission. It was never used, and is unlikely to be brought back from the dead now.
So after 50 years of space rockets, we're left with the old Proton and the Ariane 5, and a slightly improved Atlas based on Russian engines, as the heavy ones. Not exactly progress, from the point of view of putting large interesting things in space.
So what else was available, besides chemical rockets? Well, parallel with Saturn V development the US investigated nuclear rockets. A mostly workable nuclear rocket, NERVA, was eventually produced, and even considered for the top stage of the Saturn V, and various future heavy rockets. But it had issues with the reactor being damaged by vibration, and by the time these were sorted out, the public was increasingly unhappy with the idea of nuclear rockets. So out that went.
Next, there was Project Orion, a study conducted for the US in the 50s and early 60s by a company revelling in the name "General Atomics". This was basically a craft propelled into the air by nuclear explosives. No, really. Cost per kilo? 70cents modern money. Compares very well to modern solutions. This was based on a craft with a diameter of 400m, massing 8,000,000 tonnes; smaller craft would be somewhat more expensive. Equivalent fallout to a 10megaton bomb, though this could potentially be reduced or eliminated using clean fusion bombs. A small chemical trial was conducted. Ultimately, of course, the whole thing died of politics.
Then there are great big guns. Iraq started building two ("Babylon Guns") before the first gulf war; they would have been able to fire a few tons at a time into orbit, for about twenty times cheaper than rocket. A scale model (300mm instead of 1000mm) was completed. Various people are currently looking at building something similar.
Okay, that's enough of that, for now :). Silly rockets.