Reserves held by commercial banks are really just current account balances at the central bank, akin to current accounts of individuals at commercial banks.
Most banking transactions between customers of different banks are ultimately settled through transfers between reserve accounts at the central bank. Technically, it is possible for commercial banks to perform such transactions without a central bank, but this would require correspondent accounts between the respective commercial banks. Additionally, it exposes the banks to credit risk. Therefore, centralized payments via central banks are the preferred mechanism for settlement between commercial banks.
While any individual commercial bank is free to choose between reserves and other assets, the system wide quantity of reserves is determined by accounting identities on the central bank’s balance sheet only. If an individual bank decides to add reserves, it can reach out to the central bank and engage in a few transactions, the most common being a REPO: a sale and repurchase agreement where an asset is sold to the central bank (frequently government bonds, but also eligible loans etc) in exchange for central bank reserves, while simultaneously agreeing to repurchase the asset for a predetermined price on a specific future date. On the commercial banks balance sheet, you have the additional reserves as assets, and the repo as a liability.
To understand the way system wide reserves work, it helps to think of a model economy with only two banks. A new bank (call it N), and an old bank (O). If someone decides to deposit at N, they must have had the money at O beforehand. Therefore, deposits transfer from bank O to N, which simultaneously requires that reserves move from O to N. This implies that there is no impact on the total quantity of reserves.
Edit A repo still adds reserves. Additionally, some REPOs are very long term, e.g. TLTROs which were immediatly deposited as reserves by banks. In the end, reserves change all the time. See for example the official series for total reserve balances maintained plus vault cash used to satisfy required reserves on FRED.
A pivotal element is that if a central bank does not provide or
allow extra reserves in the system, then any increase in holdings
must come at the expense of another commercial banks reserves. However, modern central banking largely operates under an ample reserves framework, where reserves are supplied in amounts that leave most commercial banks with reserve balances well above what is required for payments purposes. The TLTRO example above is such a system where banks hold large amounts of excess liquity (ECB: Balance sheet size and
interest rate control).