Visitors who plan on visiting Taiwan for more than 30 days and less than six months, or are ineligible for Visa-Exempt Entry for less than 30 days, should apply for a Visitor Visa. Visas are issued for the following reasons: transit, tourism, visiting relatives, attending a training course, receiving medical treatment or engaging in business or other permissible activities.
Some requirements apply. The passport must be valid for at least six month, two photos must be supplied with the application form, an outbound ticket must be presented (or steamship ticket, if you can find a steamship) and also possibly documents verifying the purpose of the visit.
Visas are issued for 14, 30, 60 or 90 days - depending on nationality. Some visas are extendable, some are not. Some visas are single-entry only, and others are for multiple entries. Fees also vary by nationality.
For the latest details, and details specific to your nationality, please check out the Ministry of Foreign Affairs website (see below). Visa application forms may also be downloaded from their site.

Useful Links
MOFA, Bureau of Consular Affairs - Visa information