On the vast majority of banknotes, it doesn't mean much of anything anymore. Historically, banknotes were redeemable (at times practically, at other times only theoretically) for a specific amount of precious metal (usually gold). But that hasn't been true for most currency in decades. For example, the Bank of England has this to say about the phrase on their banknotes:
> The words "I promise to pay the bearer on demand the sum of five [ten/twenty/fifty] pounds" date from long ago when our notes represented deposits of gold. At that time, a member of the public could exchange one of our banknotes for gold to the same value. For example, a £5 note could be exchanged for five gold coins, called sovereigns. But the value of the pound has not been linked to gold for many years, so the meaning of the promise to pay has changed. Exchange into gold is no longer possible and Bank of England notes can only be exchanged for other Bank of England notes of the same face value. Public trust in the pound is now maintained by the operation of monetary policy, the objective of which is price stability.
We need to think back to what notes were actually used for. Early on, currencies were tied to something, usually to the value of gold, also known as the "gold standard", or silver, known as the "silver standard". What this actually means is that your one dollar note was worth a certain amount of gold. This meant that you didn't have to carry gold coins around, which was of course handy. At that time, you could have taken the bill to a bank and demanded they give you the corresponding amount of gold (or silver, respectively), and they would have been obligated to do so. Conversely, you could have taken an amount of gold to the bank and exchanged it for the corresponding amount of dollars. Nowadays that is mostly not the case, as most currencies are not tied to any valuable material or other currencies, but are instead floating, ie. their value is determined by how much people are willing to pay for it. The text mostly remains in bills that are still tied to something, be that gold, silver or even another currency (it would be possible to tie the value of a currency to the US dollar, for example, in which case you could demand US dollars from the bank), but in floating currencies it is either omitted or a relic from that age.
It's generally old text that no longer means anything. Originally, bank notes were issued by a bank to represent *real* money (e.g. gold) the bank had, but which wasn't really suitable to carry around all the time. Instead of physically giving you actual gold for a purchase, I'd give you an IOU from a bank saying "we owe whoever has this note $X in gold." But you probably didn't want to deal with physical gold either, so you could just hang on to that something and give it to someone else instead of giving them physical gold. At any point, someone could take this IOU to the bank and get it converted to real money.
Over time, these IOUs became just as much "real" money as the gold was; there was generally no reason to convert them to gold. Laws were passed making it so that they counted as a legitimate way to pay off a debt, which gave them a second thing (besides getting gold) that they were legally guaranteed to be useful for. Since exchanging them for gold no longer served much purpose and made it harder to keep the economy running smoothly, governments dropped the idea that they just represented real money. At this point, banknotes *are* real money, not a promise of a certain amount of a commodity. But some designs might still keep the old text.