Taiwan is one of the few countries in the world with completely free, nation-wide wifi. It's based in hotspots around the major cities, such as bus stops, MRT stations, even the occasional 7/11. I've found that even buses sometimes have some sort of cellular connection to this wifi network. Across Taiwan, the network is known as iTaiwan. In Taipei, it's known as TPE Free.
Registration is fairly straightforward. The wifi connection points (named either iTaiwan or TPE Free) are not password protected, allowing you to connect at any time with any device. However, your connection will not allow access to the internet. Attempting to access the internet cause a redirect to a login page for the iTaiwan/TPE Free network. Before being able to use the internet through iTaiwan/TPE Free, you need to register with the network.
Registration can be done overseas from certain countries. If you have a mobile number based in Japan, Singapore, Thailand, India, Vietnam, Turkey, Germany, Switzerland, Italy, Spain, Greece, France, USA, Mexico, Canada, Australia, New Zealand, South Africa, Egypt, or Saudi-Arabia, you can register using your phone number before coming to Taiwan. If you're already in Taiwan, obviously a Taiwanese phone number will suffice as well. If you're in Taiwan, but don't have a Taiwanese phone number, I will address another method shortly.
By visiting this link you will be presented with a form through which you can create an account. Simply put in your mobile number, your desired password (make this seriously easy to remember. You're gonna be SOL if you're lost in an MRT later and can't remember your password), and your desired email address, which will double as your login ID. After going through these steps, you should get a verification code. Your success rate may vary. I tried with a google voice number and it didn't work, but don't lose hope, if it doesn't work, there's another way. In any case, after getting the verification code, enter it, click through some more confirmations, and you will be officially registered for the network. Skip to "logging into iTaiwan/TPE Free" if you were successful.
If for some reason that doesn't work out for you, try the following:
Registering at a Counter
If the phone options doesn't work for you, you can register at one of several "counters" throughout the city. The locations are Taipei Songshan Airport, Taipei Man Station, "all major MRT stations," and "tourist attraction sites," as well as many hotels throughout the country. I would bank on Taipei main station. You'll need your passport and email, but don't fear, you don't need email access to activate.
Here is the location of all of the "counters."
Here is the location of all the hotels.
The people at the counter generally have a decent grasp of English, and considering that nobody talks to them ever, they are practically desperate to help you. Expect 5-6 people in uniform with enormous smiles talking over each other in excitement over finally having a problem to solve. Get to the counter and they will take care of you.
Logging in is a little bit counter-intuitive, but quite simple. The important thing to remember is that the wifi is finicky, and you may need to move around in whatever MRT you find yourself in in order to get a strong enough connection to actually be able to access the login page. In any case, here is a visual guide for TPE free. Basically, use your device's wifi service to connect to "TPE Free" or "iTaiwan." Then, open your browser of choice and attempt to access "google.com," or whatever website. You'll be presented with a login page. Input your email/telephone/loginID you chose earlier, along with your password, hit "login" or "註冊." If you're not sure, just avoid hitting any buttons with "不" and try to find "同" (不同意 means "don't agree) and "同意" means agree). Be patient, you'll be flipped quickly through several different pages as the network does its thing. When you see "Redirecting..." it's at the final step. It'll land you at a final launch page and you should then have total access to the internet through your browser and other apps.
The wifi is pervasive, but far from universal. Mostly, count on it being available at all MRT stations in Taipei. Your results will vary greatly outside the city. Sometimes you can get lucky on a bus or near a 7/11, but I've found these locations to work only rarely. Here is a search tool for every single location, but, like the ubike search tool at the kiosks, this is pretty much useless for a foreigner. There is apparently an app for iDevices and android devices, so you can give that a shot, although whether it works offline is a toss up. The most useful would probably be this overlayed google map, which, amusingly, if you zoom out, shows that the entire city of Taipei is seemingly completely covered by hotspots.
In any case, when in doubt, just find your nearest MRT station.
I've encountered several bugs, and my solutions are as follows:
"Connected" to the wifi point, but unable to access the internet: Could be that your connection is too weak. If you're in an MRT station, try going to another spot. Also, try resetting your wifi antenna, or your device.
Was connected at one MRT station, but not another: You're gonna have to re-login. Hopefully your browser has a "save login information" setting, so you don't have to manually input it every time.
Comment if you encounter any other bugs!
Map of Wifi Hotspots in Taipei: http://www.tpe-free.taipei.gov.tw/tpe/hotspot_map_en.aspx
Search Tool of All Wifi Hotspots: http://www.tpe-free.taipei.gov.tw/tpe/Hotspot_en.aspx
Wifi Location App: http://apps.taipei.gov.tw/ct.asp?mp=100091&ctNode=50155&xItem=48919185
Guide for Loggin In: http://www.tpe-free.taipei.gov.tw/tpe/EN-TPE-Guide.html
Location of all Tourist Counters: http://www.taipeitravel.net/frontsite/en/cms/cmsAction.do?method=goCMSDetail&contentId=295&menuId=1030202
General Website for TPE Free: http://www.tpe-free.taipei.gov.tw/tpe/index_en.aspx
General Website for iTaiwan: http://itaiwan.gov.tw/en/faq_service.php