Brain drain is the loss of a country's best and brightest people due to greater work opportunities in more developed countries. This is a crippling problem in most Third World countries, as the local economy loses the skilled workforce it needs. Unlike brain drain from First World countries like Canada, it isn't "massively amusing" - it is a very real problem.
Several causes of brain drain are higher (monetary) opportunities in the US or Europe, where a college-degree holder can make ten or twenty times as much as back home. Other reasons to emigrate are religious persecution, political persecution, or being fed up with rampant corruption and lawlessness in the home country. Brain drain from Third World countries doesn't just involve people working abroad and retiring at home - in many cases, they aim to transplant their entire family into a better environment - i.e. in another country.
Some countries, like Korea, are able to reverse this trend, and are consequently able to build the local high-tech industries using returning engineers and scientists educated abroad. Others, like the Philippines, experience worsening brain drain, as locals dream of nothing better than being able to leave the "hellhole" (a common term for these countries, as in "what are you still doing in that hellhole?"). School-teachers going to Singapore or HK to work as domestic help are the norm, not the exception. Even prostitutes prefer to work in Japan or Brunei.
It's gotten so bad that the Philippine government has basically given up on trying to fight brain drain, and has resigned itself to cutting itself a piece of the OCW (Overseas Contract Worker) pie. In fact, OCWs contribute billions of dollars yearly to the local economy. Unfortunately, the ultra-rich give those billions back by buying condos in New York or LA, or amassing huge amounts in Swiss banks.
I've taught Computer Science in college for eight years, and I've tried to convince my students to stay and make a difference. It's pretty hard, especially if you realize that by getting even an entry-level job as a maid or nanny abroad, he or she can feed his entire family (and in many cases, the families of all their relatives). There's a feeling of helplessness every time one of my friends comes home for a visit and asks why I'm still here, when with my intelligence, I could be making millions in the US. The old "I'm doing it for my country" line doesn't cut it anymore.
Anark: When brain drain occurs from totalitarian states, we call them boat people. Those wet, hungry, freezing people paid hundreds of dollars to smugglers to get them on those boats - they're usually middle-class (or even upper-class) people back in their home country.
I've talked to several Koreans - many came home because they are proud to be Korean, and wanted to use their knowledge to rebuild their country (after WWII and the Korean War). We don't have that. Very few Filipinos are that nationalistic.
Of course, what you describe in Mid-East states as throwing money at the problem works pretty well, too.
Yes, we have academic grants. But what usually happens is the bright kid gets to the US, finishes college, gets a high-paying job, and breaks his or her contract with the home government. Most don't even bother paying off the debt. Third World countries can't retain either quality nor quantity. Any worker sufficiently intelligent to get a job abroad is also sufficiently intelligent to get around any "leashes" the government can put on them, since most banana republics aren't renowned for their competence.
And yes, I'm free to leave any time. I keep my passport and a valid visa ready for a quick getaway. And after over twenty-five years of dealing with corruption, bureaucracy and greed, I'm about ready to call it quits.