When a person wishes to obtain shares in a company in return for cash, this is known as a 'subscription'.
This document is a letter agreement by a single individual or corporate subscriber under which they agree to pay (or 'subscribe') for shares in a company in return for a cash payment. It sets out the price that the subscriber has agreed to pay for each shares to be issued to them and the total amount that they agree to pay. It also authorises a company to enter the subscriber's details into the register of members upon receipt of payment for the shares.
Although such an agreement is not strictly required for someone to be issued with shares in a company, it avoids any confusion over how much is being paid and how many shares are being issued in return.
You would use this document when issuing shares to an individual or company. You shouldn't use this document in relation to a transfer of existing shares from one shareholder to another. The document assumes that no warranties (promises) are required to be given by the company or any of its shareholders (typically the founders or majority shareholders).