A lot of things happen at the moment the fuel is injected. If the fuel was somehow perfectly microscopically atomised on injection, it would not self-ignite at all. It would have trouble igniting even with a spark plug. Ignition needs a richer fuel-air mixture, and this is locally provided by the start of the injection being full of larger drops of fuel, because it occurs at a lower than maximum pump pressure (about 2-3 times lower). The evaporation of components from the fuel as it enters the hot compressed air is what makes it ignite. This is called creation of ignition precursors, and it is responsible for a delay between injection and ignition. Once this happens, the temperature rises sufficiently for the rest of the injected fuel to ignite. This is the principle behind pilot injection, a technique that makes things like the HDi common rail possible and quiet compared to old style direct injection engines.
Incidentally, the ability to reliably create precursors with consistent timing is what the often misunderstood cetane number of diesel fuels is all about. The only reason it is measured is because this number and the timing accuracy strongly correlate. The higher the cetane number, the more consistent is the precursor creation across a varying range of temperatures and amounts of fuel injected. This is also why diesels misfire and sound much louder on bad quality fuel.
The cetane number is the measure of the volatility of the fuel. If this volatility is higher at the conditions present at injection time, the time from injection to combustion is shorter--and time is a valuable comodity in a diesel engine. Besides, self-ignition is a statistical event, the time interval elapsed between injection and ignition has properties of a chaotic event. The shorter the time, the narrower the statistical dispersion of the time of ignition, and the better controlled the engine timing. In practice, higher cetane fuels misfire less and result in a quieter engine. We would want both high cetane and octane at the same time but, unfortunately, they are mutually exclusive to an extent. Winter diesel, for instance, has a higher cetane number than summer one. There are some additives that improve but they are usually not worth their price, especially with Euronorm diesel.
As you can see, correct timing is very much chemistry dependent on diesels. This is why really accurate timing requires feedback using a knock sensor on the engine. Normal diesel strobes use injection time as a reference, and it is amazing how far they can be off for substandard fuel.
In theory, the fuel is always injected into the cylinder, so all of the diesels are, in a way, direct injection. In a similar manner, all gasoline injection engines except the newest GDIs are not really injection at all because the fuel is not injected into the cylinder, rather into the intake manifold--thermodynamically speaking that's not part of the engine. But that's a different story.
DI or direct injection occurs into a combustion chamber that is a part of the piston-cylinder combination (on small engines, it's a toroidal excavation in the face of the piston). II or indirect injection occurs into a combustion chamber which is part of the head, and is connceted to the cylinder through an orifice.
Why the difference? Well, they are really two different compromise solutions to the same problem--that the fuel does not burn up immediately. The combustion chamber has to be relatively small and swirl-shaped to ensure good combustion occuring away from the chamber walls, remaining that way until combustion is over. Short of a 5-stroke engine, this is not achievable. Indirect injection solves the problem of a controlled chamber in a simple way: the orifice creates high swirl in the small chamber as the piston compresses air in the cylinder and it rushes through the orifice. There are several types of chambers but only three are in widespread use, one is a toroidal swirl vertical chamber (patented by Mercedes), the second is a turbulance chamber (use by everyone else), and the third, a new, solitary and very late arrival is the spiral swirl chamber in the Fiat 1.9TD engine that replaced the old 1.9TD and already replaced by the 1.9JTD, after only being in service for about two years.
The solution also works backwards: the pressure drop in the chamber as the piston goes down is limited by the orifice, prolonging the time available for fuel injection and burning. The constriction in gas flow also makes it quieter; the small explosion acts onto a limited surface area. The disadvantage is lower efficiency (lost energy because of the gases passing through a constriction) and increased fuel consumption, and with turbocharging, the engine is required to give higher output with the same size prechamber, increasing the demands on the material of the prechamber (cracks in the prechamber orifice are a typical problem even on some newer II turbos).
Direct injection solves the problem by closing up the chamber when the piston is in the upper position, opening it up again as the gases expand and push the piston down. As a consequence, the chamber is initially the size of the whole cylinder, and swirl due to gas compression happens only as the piston reaches the very top. Once it's there, the time for injection and combustion is very short. And, when it happens, the chamber expands again, making the explosion act on a very big surface area, with a lot of fuel involved at once--hence the noise. However, there is no constriction, efficiency is excellent with low fuel consumption. Because the chamber is present only when the piston is in the top position, to have enough time for injection and combustion, these engines have been limited to low revs (all big diesels are DI). For small engines, the combination of low starting revs and requirement for good swirl (better at high revs) makes for difficult starting.
However, some new developments over the last decade made this principle practical even in higer rev, smaller engines:
Turbocharging plus accurate modeling of intake manifolds, channels, and valves. This enables the inward rush of the air as the intake valve opens, to create the initial swirl. Such modeling requires considerable computing power, and is still really in developemental phase; but it is nice to know that even with the advanced DI engines of today there is ample room for improvement...
High pressure injection systems shorten the time needed to inject the fuel, giving more time for combustion. In addition, injectors are multi-jet enabling combustion to start at several places at the same time, in effect sectoring out the combustion chamber to let the combustion to occur in several sectors in parallel. This is incidentally the same principle used on twin-spark gasoline engines.
Better control of the injection process by advances in pump and injector technology. This is essential because with the shortened interval for everything to happen, timing related error magnitudes are also smaller. A side developement of this is pilot injection: the starting drop of fuel, producing a sudden rise in pressure, temperature, and turbulence, makes the rest of the injected fuel burn almost immediately, uniformly and quickly.