ATMs and CDMs

ATM (Automatic Teller Machine) - 提款機 (tíkuǎnjī)
CDM (Cash Deposit Machine) - 存款機 (cúnkuǎnjī)
ATMs in Taiwan are ubiquitous. You'll find 24-hour ATMs in most convenience stores and banks, and there are also plenty of ATMs scattered throughout department stores, MRT and train stations, airports, etc.
While the majority of ATMs in Taiwan now accept international cards, some ATMs accept only local cards, so be sure to check the ATM for stickers (VISA, PLUS, CIRRUS, etc.) before trying. If you have a local card, you'll notice it has both a chip and a magnetic stripe. Europeans are probably familiar with this, although chips are not yet popular in North America. You'll need to set a 6-digit PIN for your chip (used for local ATM withdrawals) and a 4-digit PIN for the magnetic stripe, if your card can be used outside of Taiwan. If you don't set the 4-digit magnetic stripe PIN, you may find yourself with an unusable bank card when you leave Taiwan.
You'll also find it convenient to make sure your bank has CDMs as well, and some banks even have ATM/CDM combos. CDMs are much less commonplace than ATMs. You can use them to deposit cash directly into your own bank account or another bank account.
CDM (Cash Deposit Machine) - 存款機 (cúnkuǎnjī)
ATMs in Taiwan are ubiquitous. You'll find 24-hour ATMs in most convenience stores and banks, and there are also plenty of ATMs scattered throughout department stores, MRT and train stations, airports, etc.
While the majority of ATMs in Taiwan now accept international cards, some ATMs accept only local cards, so be sure to check the ATM for stickers (VISA, PLUS, CIRRUS, etc.) before trying. If you have a local card, you'll notice it has both a chip and a magnetic stripe. Europeans are probably familiar with this, although chips are not yet popular in North America. You'll need to set a 6-digit PIN for your chip (used for local ATM withdrawals) and a 4-digit PIN for the magnetic stripe, if your card can be used outside of Taiwan. If you don't set the 4-digit magnetic stripe PIN, you may find yourself with an unusable bank card when you leave Taiwan.
You'll also find it convenient to make sure your bank has CDMs as well, and some banks even have ATM/CDM combos. CDMs are much less commonplace than ATMs. You can use them to deposit cash directly into your own bank account or another bank account.
Using an ATM
ATMs here are mainly identical to those you're familiar with in your home country, with a few notable exceptions. For foreign cards, you'll be limited to cash withdrawals (and possibly balance check). For local cards, you can make a bank transfer from any ATM to any other local bank account at any local bank.
For withdrawals, you'll often be asked if you want to receive 100 NT notes or 1,000 NT notes only. If you choose 100 NT notes, you'll get 10 x 100 NT and the rest of your withdrawal in 1,000 NT notes. The English label on this option is often labelled something confusing like "Change 100?" Just look for the "No" or "1,000" option on the screen.
After selecting your amount and note type, there is always a screen asking if you want to continue with another transaction after the current transaction is completed. The English for this screen is almost always nonsense, usually something like "Do you want to continue?" with the options "Continue" and "Cancel." You'll want to hit "Cancel" or "No" on this screen to get your card back, even if the English on the machine leads you to believe you will cancel your current transaction.
For withdrawals, you'll often be asked if you want to receive 100 NT notes or 1,000 NT notes only. If you choose 100 NT notes, you'll get 10 x 100 NT and the rest of your withdrawal in 1,000 NT notes. The English label on this option is often labelled something confusing like "Change 100?" Just look for the "No" or "1,000" option on the screen.
After selecting your amount and note type, there is always a screen asking if you want to continue with another transaction after the current transaction is completed. The English for this screen is almost always nonsense, usually something like "Do you want to continue?" with the options "Continue" and "Cancel." You'll want to hit "Cancel" or "No" on this screen to get your card back, even if the English on the machine leads you to believe you will cancel your current transaction.