As a foreign student coming to Taiwan for a Master's degree, you are eligible to apply for a "Resident Visa for Foreign Students".
There are two ways to do this, you can either 1) come to Taiwan on a visitor visa first, then switch to a resident visa here in Taiwan at least 7 days befor your visitor visa expires, or 2) you apply for a resident visa through a Taiwan overseas mission center at your home country (recommended). We recommend the second approach since if the overseas mission center would like to see some additional documents, it's much easier to obtain them while you are still at home.
Below we provide a brief overview of the procedure, but for more details please check these two links below:
Here are the procedures for visitor's visa application: