How can a citizen of Mainland China visit Taiwan?

You can't get a visa per se, since Taiwan, ROC and China are considered by both governments to be the same country, but you may be able to get an entry permit (which, of course, is a defacto visa). The requirements are listed in detail at this website. In part, badly google translated below (for 'job applicant' = employed applicants- students, unemployed and retired applicants have different rules). I believe permission is also required from the mainland authorities for PRC residents.

1. The job applicant

1) proof of service: work unit issued proof of service 1 (the rise of paper >printing proofs attached), or a valid work permit one copy.

2) business license: where the company business license (for an official cover) >1. The case of foreign-funded enterprises also need foreign investment approval >certificate.

3) Application Form: I confirm autographed "mainland people to Taiwan for >sightseeing application" 1.

4) Photos: 6 months white bareheaded 2-inch color photo of 3, should be held by >residents of mainland China ID card can be identified as the same person.

5) ID card, household: copy of ID card (positive and negative multiplexed on the >same sheet). 1 copy of the account.

6) If both husband and traveling together: One copy of the marriage certificate.

7) Evidence of assets: real estate license or passbook (above $ 50,000) or other >proof of assets one copy.

8) mainland residents to Taiwan Pass: mainland residents to Taiwan passport copy >1 (must be valid for six months or more).

These things tend to change frequently, so it may be different now or in the near future, especially with the new government of Taiwan (province).

Last I looked there were a few offices (defacto consular offices) in key mainland cities.

Wikipedia offers the following somewhat confusing entry:

PRC citizens with mainland residency only

Although the ROC does not issue visas to PRC passport holders (except for mainland residents who have resided in foreign countries for at least four years and have legally acquired the nationality of a foreign country[3]), mainland residents may first apply for a Mainland residents Taiwan Pass (Simplified Chinese: ???????????) from PRC authorities[4][5][6] and then apply for an Exit and Entry Permit Taiwan Republic of China (Traditional Chinese: ??????????????) from ROC authorities. Taiwan generally requires that these applicants travel with a tour group, travel individually if reside in cities under Individual Visit Scheme, be invited, and/or provide a 200 000 TWD (50 000 RMB) deposit.[7]

PRC citizens with mainland residency, who have permanent residency abroad or in Hong Kong or Macau, or who are studying or have work authorization abroad or in Hong Kong or Macau, are not required to get a Mainland residents Taiwan Pass; instead, they provide the proof of their relevant foreign status when applying.[8]

If you are interested in visiting Taiwan for only upto 3 days, it should be possible to do this without visa as long as you have a connecting flight to a different country.