How does one get and where does one apply for a passport from another country? My country's passport doesn't give much visa-free entry. I heard getting a passport from another country open doors to travellers.
To acquire a passport from another country, you need to be a citizen of that country.
If you are already a citizen of a different country (which can sometimes be the case even without your knowledge, but still has to be claimed to benefit from being one), you should be able to find some information on the legal procedures to follow online. Most commonly in such cases, you will have to deal with a consulate of that country in your area.
If you are not a citizen of any other country, you're mostly out of luck. Every country has its own citizenship laws, and there's no universal recipe to the simpliest way to acquire a second citizenship, it depends on your personal situation.
However, if you happen to be rich, you might consider legally purchasing citizenship from such places as St. Kitts & Nevis, St. Lucia and Vanuatu (again, you can find more of these places and more info about them on the internet). The pricing commonly starts around $120,000 and can go all the way up to $1,000,000 and above. As you may have noticed, all (or most, if speaking generally) of these places are islands, quite a few of which have struggled through devastating events such as hurricanes. This is why these islands are selling citizenship, this is their way of rebuilding their economies.
You normally need to be a citizen of a country to get a passport. To do that you would, in most cases, have to live in the country for some time first before you can qualify to become a citizen.
As others have said, you first have to be a citizen of the country whose passport you wish to have. Citizenship can be acquired by
- By Birth (Jus soli) - by soil. This is given who babies who are born in the country. Some countries allow unrestricted Jus soli, some insist that atleast one of the parents are legally resident at the time, some don't allow it at all.
- By Birth (Jus sanguinis) - by decent. This is given to a baby, because their parent(s), or an ancestor, had that citizenship. Some countries allow unrestricted Jus sanguinis, some require each generation be registered before the next one's birth, some don't allow it at all.
- By naturalization. Most counties will confer citizenship on a foreigner of good standing who has been legally resident (and working) in that country for the previous 5-10 years. The period is sometimes reduced if the person is married to a citizen.
- By investment. Some (mainly smaller or poor countries) allow persons to invest a sizeable amount of money to the country, and in return they will be given citizenship.
If you are interested in gaining a second citizenship, then you should check the citizenship(s) and birth-places of you, your parents, and your grandparents. Some countries do allow this to go further back, including Germany (anyone whose ancestors were expelled by the Nazis), Italy (anyone who has an ancestor in the male line born in Italy since the formation of Italy in 1861), Spain (anyone who can prove that an ancestor was expelled from Iberia due to the Moorish invasion of 756-768). There are others, but you'll need to research your family tree, and then check each countries citizenship laws.
Also, some countries allow Jus soli and Jus sanguinis to be granted to persons (or ancestors) who weren't born in that country, but in region that that country holds a territorial claim to. For example, Ireland treats Northern Ireland (part of the UK), as part of Ireland for citizenship reasons.
There can be consequences of gaining second citizenship, especially if the country has compulsory military service - you should check that before applying.
@user4551's answer covers most of it, but there are a few countries (Italy, Ireland, and Lithuania come to mind) where you can claim citizenship and hence get a passport if your grandparents or parents (and sometime great-grandparents) were citizens of that country. A Google search should help you find out if there are any other countries like that, but here's one example for Italy: Italian Consulate of San Francisco.