Power of Attorney (PoA), also known as the Letter of Attorney, is a form or document that you can use to legally empower a third party to act on your behalf. In other words, by signing a Power of Attorney, you are allowing someone else to make a decision for you or represent you in front of the authorities. Power of Attorney is also used in IP law, and you will most likely need to sign one when registering a trademark.
Practically speaking, signing the PoA with an attorney or law firm allows them to represent you during the case itself, as well as carry out any follow-up actions (such as filing a cease & desist letter, responding to the office action or opposition) if needed.
Not every IP Office requires the Power of Attorney to be signed, but it is usually mandatory as it protects both sides. To proceed with filing the application and finishing the trademark registration process, both parties must sign the document (PoA), which will be presented to the IP Office.