A promissory estoppel is a legal term which refers to a set of various rules aimed at preventing people from holding out on promises they have made, when it's related to a legal case. The doctrine basically states that a person making a promise is obliged to comply to it, even if they had made the promise without any serious consideration for its consequences. However, in order for a promissory estoppel to be upheld in court, several circumstances must be proven by the other party - including their reliance on that promise being kept, and proving that the promise was made in a serious manner and not, for example, as an obvious joke.
There are various examples of this, but it's most commonly used to refer to verbal promises and not written ones - due to the fact that written promises are universally regarded as contracts and different sets of laws apply to them. But, for example, if a party A makes a verbal promise to party B that they're going to provide them with some specific information and party B relies on that information for their personal reasons, party A can be upheld with a promissory estoppel which obliges them to provide that information regardless of the consequences it may have for them.
Due to the flexible nature of the term, it's one of the definitions most commonly exploited by individuals, which is attributed by the fact that it can be very difficult to prove that a promise has been made when it has not been written down.