Your passport is issued by your government and provides proof of your identity and citizenship. A visa is issued by a foreign government and gives you permission to gain entry (or exit) to the country, usually for a specific period of time. There are many variations on visas.
Most, but not all, countries require an entry visa. Exit visas used to be important but are required by fewer countries now. The paperwork required to get a visa vary widely across countries. In some countries, a visa can be obtained at the airport pro-forma. In others, getting one is a long, complex, and costly affair. A visa is a privilege, not a right, and any country can deny one for a variety of reasons, with or without explanation.
There are different types of visas for different purposes. Common types include a tourist visa, a business visa, student visas, and residence visas. Each has its own requirements and restrictions. Visas can be for “single entry”, a specific number of entrys (2, 4, etc.), or “multiple entry”. There is usually a limited time (such as 30 or 90 days) within which your initial visit must be made, and they can be good for anywhere up to several years. In some cases, it is possible to request more entries or a longer time period, but the details of what is actually issued often seems arbitrary, and is frequently not – as a practical matter – appealable.
Visas often require a letter of invitation. This may require specific information depending upon the country. Most often, it is on the sponsoring organization’s letterhead, is addressed to the government embassy or visa issuing authority, identifies the applicant, and requests a visa. The sponsoring organization must state that they assume financial responsibility for the individual while they are in the country, and for their transportation back to their home country.
When you have all of this paperwork assembled, you send it, money, and photos off to the embassy of the country for which you wish a visa. You have to include your passport & in some countries your yellow card. Seeing your passport and yellow card disappear into the postal mail can be a real gut-check. You’re at the mercy of the worldwide bureaucracy. Get used to it.