So first thing we're going to touch on is what "stealth" is and isn't. Stealth is vernacular for passive means to reduce an aircraft's visibility to radar, it does not mean totally invisible to radar. Each aircraft has what is called a radar cross section (RCS), ie how much energy is reflected off the aircraft back at the receiver (radar). The larger the radar cross section, the easier it is to detect and aircraft. Usually the "size" of a radar cross section is expressed in meters^2. Pointing a radar head on to an F-117, the reported RCS is .001-.003m^2, for comparison an aircraft like a B-52 has a RCS of 100m^2, and an F-15 (similarly sized to the F-117) has a radar cross section of 25m^2. The F-117 has such a small RCS because of its unique geometric shape (the classic arrowhead/ angular shape of the air-frame) and the use of radar absorbing materials. An easy way to think of the advantages of stealth is it gives the defenders less time to spot the attacking aircraft and destroy it, eg: a hypothetical air defense battery can engage an F-15 at 200 miles giving them 200 miles to destroy the aircraft, but they can only spot an F-117 at 25 miles giving them less time and space to destroy the attacker.

As for the shoot down itself, on the night of March 27, 1999 the Serbian 250th Air Defense Missile Brigade under the command of Lt Col. Zoltán Dani fired two S-125M (SA-3) missiles at an F-117 piloted by Lt. Col. Dale Zelko. The first missile missed, but the second missile successfully found its target and downed the stealth fighter. How a tiny nation using outdated weapons could knock down one of the world's most advanced combat aircraft is as much a product of the smart tactical thinking of the Serbians and of Lt Col. Dani, as it was the limits of stealth technology.

The F-117's angular shape is a product of the limits of 1970s computing and required it to fly pre-planned flight patterns to maximize the effectiveness of its radar reflecting shape. Flying out of bases in Italy the F-117s flew regular and predictable flight patterns to strike targets in Serbia. It has also been reported that the Serbians had been able to intercept communications between NATO air traffic controllers and NATO strike aircraft, effectively giving them a clearer "picture" of the air-space. Lt Col. Dani decided to set his surface to air missiles in what was effectively an ambush of the American jets. Previously Serbian air defenses had reportedly noticed that at close range low frequency radars, such as the P-18 (Spoon Rest D), could track F-117s within 15 miles when tuned to their lowest frequency. The P-18 is a long range surveillance radar, meant to track aircraft at long range, it is not a targeting radar (what you use to get a missile "lock") which are typically high frequency radars. On the night of the 27th Dani having pre-positioned his missile battery used his P-18 low frequency surveillance radars to approximate where the F-117s were and then directed his high frequency targeting radars in their general direction. He then ordered the targeting radars for his S-125Ms to turn on in a twenty second burst to try and acquire a target lock, but they failed to find anything (having studied the fate of previous air defenses who opposed the US, Lt Col Dani had learned not to leave his radars on too long lest they become the target). Knowing the risk Dani ordered the targeting radars to turn on for a second time, but they again failed to find a target. Dani again ordered the radars turned on for a third time and at a distance of eight miles they locked onto Lt. Col Zelko's aircraft and fired two missiles. One theory as to why the F-117 was visible to the targeting radars was because the aircraft's bomb bay doors were open, thus giving a greater area to reflect radar energy off of and effectively increasing the "size" of the F-117's radar cross section. Either way it remains a clear demonstration of the very real limits of even the most advanced technology and what happens when you fail to anticipate your opponent thinking imaginatively.

First off, it's important to understand that "stealth" aircraft are not radar invisible, the more accurate term for such aircraft is "low observable" or (specific to radar) "low radar cross-section". Additionally, the radar cross-section of a "stealth" aircraft depends on several factors such as radar wavelength and bandwidth, relative orientation between the aircraft and radar site, and aircraft configuration. On top of all that you need to factor in the battlespace and external conditions. For example, most of the time when a stealth aircraft operates it is accompanied by signal jamming vehicles (during Kosovo this would have been EA-6b Prowlers) as well as a contingent of aircraft with beam following missiles anti-radar station (HARM) missiles. Adding all of that together means that the "stealth" aircraft would be difficult (but not impossible) to track via radar, even more difficult to get a target lock for a SAM station, and any operating SAM station would be inviting its own destruction merely by staying on long enough to track and get a lock on an aircraft.

There are several factors that went into making the shootdown possible, some of them due to mistakes on the part of the US armed forces and some of them due to cleverness on the part of the Serbs, along with a little bit of luck.

The major mistake that made the whole thing possible was that the US were flying sorties day after day at the same time using the same flight path. Which gave the Serbs ample opportunity to study the flight paths very closely, observe where the planes flew (visually sometimes) and even identify at which point in the flight paths the planes would be most observable (with their bomb bay doors open, massively increasing their radar cross-section) and thus most vulnerable.

The other side of the coin was the operational and technical cunning of the Serbian commander (Colonel Zoltan Dani). The key problem here is that in order to track and obtain a target lock on a plane you have to operate your SAM site's radar for a considerable period of time, which opens you up to counter-attack from HARM missiles homing in on those radar stations. First, Dani positioned his radars and SAM systems in the best possible location to hit the F-117s as they were returning to Italy after having just dropped their bombs, ideally when their bomb bay doors were still open. Second, he drilled his SAM units to be able to move their otherwise non-mobile stations in under an hour and a half, to only activate their targeting radars for at most 20 seconds at a time, and to move after every single time they had their radar on, regardless of whether they fired or not. Together with operating several dummy SAM stations using salvaged radars (such as from abandoned jets) these techniques resulted in zero losses for Dani's SAM sites despite all the forces arrayed against him.

On top of that, Dani was able to tweak his tracking radars to have a very narrow bandwidth which would allow him to get a course location of the F-117s (within about 15-miles) along their flight paths, without being detected by NATO anti-radar receivers and warning systems. His crews would manually point the SAM station targeting radars in the direction of the F-117s so that when they turned on they would require less time to acquire a lock and could successfully fire a missile. Dani was able to follow the F-117s day after day with the low-bandwidth radar and come up with a plan that had the maximum chance of working.

Finally it was a matter of waiting until the EA-6b Prowler radar scrambling planes were grounded due to bad weather (something Dani was able to learn due to observers in Italy providing him the information). At the opportune moment he had the SAM site light up the F-117 to get a lock (in that 20 second burst mode), but it didn't work on the first try. Despite his hard and fast rule of shutting down and relocating SAM sites after every radar use he pushed his luck and went for a second and finally a third try. Finally on the third try (when the plane opened its bomb bay doors) the targeting radar got a lock and they fired missiles, one of which hit close enough to the F-117 to knock it down.

Edit: One additional thing I'll add here is that even with everything working right the radar signature of a stealth plane doesn't go away, it just becomes small. This might be so small that it is effectively invisible for a given radar system, or it might just make it look like a much smaller object. This changes the problem from one of pure observability to one of identification and differentiation. Having a radar signature of, say, a bird might make your plane harder to spot, but with smart enough signal processing if your radar station is capable of spotting objects that small then spotting "the bird" that is flying at 30k ft altitude and Mach 0.8 is an easy filtering problem. This is, of course, a huge over-simplification of the problem space. Modern stealth-aware aircraft detection systems can blend together various techniques such as adaptive filtering, low-frequency (long wavelength) radar, IR/optical/acoustic sensors, multistatic radar (separating emitters from detectors), etc. Just like every other military technology it's an ongoing game of one-upsmanship, in recent times stealth aircraft have had the upper hand, but it's questionable how long that will hold given the inherent limitations involved.