This information applies to UK residents. Visas in Asia vary from place to place, and depending on your own nationality. So while this information is intended to help, we recommend that you always speak to your travel agent to find out exactly what documents you need for where.
Many countries in Asia do not require visas for UK residents: these include Thailand, Malaysia, Singapore, the Philippines, South Korea, Japan, Hong Kong, and Kazakhstan. For these countries you'll be required to fill out a document upon arrival (these are usually handed out on the plan before you land). This indicates how long you plan to stay in the country.
You are usually limited to stay for up to 30 days, and for anything longer you will need to apply for a proper visa. This can be done in the country rather than beforehand, giving you options if your plans change.
For Kazakhstan you can only stay for 15 days without a visa. In South Korea and Japan you can stay for up to 90 days without a visa. All countries may ask to see proof of your onward or return journey.
Some countries in Asia, like Cambodia, Laos, and Indonesia offer visas on arrival, which means you don't have to worry about sorting it out before you leave home. To get a visa upon arrival you must be able to present a passport that is valid for at least 6 months, as well as proof of onward or return travel. Prices for these 3 countries vary from between £25-30.
Some countries in Asia, like China, Vietnam, India, and Burma require you to obtain a visa before you travel. Without it you will be refused entry into the country, and will probably be sent home. That can put quite a downer on your gap year. They're usually quite easy to get, requiring you to go through the proper processes with the country's embassy. Often this can be done online or by post.
In the case of China you will need to have your inbound and return journeys booked, as well as some evidence of booked accommodation. A Chinese visa is expensive, usually costing just under £100. Be aware that if you arrive in mainland China and travel to Hong Kong or Macau (where visas are not required), if you want to go back to China you will need a visa that allows multiple entries, or you'll have to buy a whole new visa.