A multinational corporation is one that produces in more than one country. It's not enough to say it "operates" in more than one country - my dad sold some design work to a guy in Australia once, but he's not a multinational corporation. Prominent multinational corporations are ones like Nike, Nestle, Microsoft and Ford. After reading that list of names you're probably thinking of the other prominent property of multinational corporations, that being that they're often maligned by just about everyone, and sometimes with good reason.
A lot of multinational corporations do their production work in Lesser Developed Countries (LDCs) or "developing economies". In developing countries the comparative standard of living is lower, so multinational corporations can pay workers less than they'd have to in the West. There are other reasons for this, the first being a usual lack of laws regarding minimum wages, unionisation and safety standards. The second is that in less developed countries there is a great deal of labour and not many jobs. Because these countries are usually based on an uncertain agrarian base, people like the stability of wage labour. The opportunity cost of being employed by one of these companies isn't too high, so people are willing to do it.
And LDCs/developing economies just love attracting these corporations into their countries. Places like India and China have a lot more capital flowing into them than out, which is beneficial to the economy. This international investment brings money to an economy and gets things going. When people are paid wages, they have to spend it somewhere, which fuels the growth of the service sector (retail outlets, hairdressers, book publishers, public houses). We call this the multiplier effect, because the money which has been ejected into the economy is spent and re-spent on things which further fuel economic growth. The wages paid in these countries aren't always massive though, because companies tend to base the high-paying stuff (R&D, high-level planning) in the West somewhere, because people there are better educated. But the net result is wealth flowing into the country, even if it's just small amounts, and this increases the standard of living.
They still have to educate and train people a bit in the poor country they're producing out of, however. This makes the people more employable by other companies and in theory increases their wages, because they're a scarce commodity. If a company decides to pull out it might leave trained people behind, which could encourage another company to move in in their place (their cost of entry would be lower because they wouldn't have to train everyone from scratch). Areas can have problems when companies pull out, especially if that company hasn't been too careful on the sustainable development front. They might have used resources up, repatriated a lot of the wealth they made, and left a big stinking pile of waste and pollution. This isn't good for the country it took place in, and the governments of LDCs have some responsibility in stopping this happening.
Another positive thing multinationals can do is build up infrastructure in the country they move into. They'll initially do this to benefit themselves but in the long-run the economy as a whole benefits because it gives the country a comparative advantage over others and hence attracts more companies in. This means more wealth flowing into the country plus the assets which the companies have left behind by paying for the infrastructure. The government might do this itself because it'll be able to tax more people and then pay for services. Of course, an increase in the taxable economy which is well-used by the government in the provision of essential services (say, some basic health care and education) can be linked very directly to an increased standard of living.
As well as the developing countries enjoying an increased standard of living, so does the developed world. Because companies have reduced their cost base by producing abroad, they have become more efficient. This means they're generating more wealth, because they're either charging the same as before and getting increased profits, or they're reducing their prices and giving other people more disposable income. This is a formula for growth and more wealth generation in the developed world.
Now we come to the problems. One is corruption, which can often be a fairly big factor when multinational corporations move into 'nasty' places. Companies aren't known for their scruples and might be perfectly willing to co-operate with dictators and help these people grow rich. Governments might not ensure the property rights of their citizens are respected. For instance, if I'm a peasant farmer living in Kenya and the government sells my plot of land to Nike, I'm screwed. Unless I can find a job in the new factory then I've got no land and no livelihood. If the government wants sustainable development it doesn't help to be corrupt, but if it comes down to short-term expediency of competing with each other to see who will sell their citizens far enough up the river, it can happen.
In fact, it's mainly just corruption. No company can 'force' people to work in their factories, and people will only do it if the opportunity cost of doing so is not too great. So if they were better farming their land (and government corruption hasn't taken their land from them), no-one is stopping them. If they weren't better off, then the company is giving them a more prosperous life. Health and safety is an issue, but an unhealthy workforce isn't good to anyone (in economics we call this "depreciation of human capital"), and in the long-run it's best for companies if the workforce have all their arms etc. Arguably the labour pool might be big enough for companies not to be as prudent as they should be on this issue (ie. they can just hire new people), to which the answer is more companies, more growth, more jobs, and higher wages!
Ain't capitalism brilliant? Answers on a postcard to the usual address