A cryptographic hash function is a hash function which takes an input (or 'message') and returns a fixed-size  string of bytes. The string is called the 'hash value', 'message digest', 'digital fingerprint', 'digest' or 'checksum'.

The ideal hash function has three main properties:
It is extremely easy to calculate a hash for any given data.
It is extremely computationally difficult to calculate an alphanumeric text that has a given hash.
It is extremely unlikely that two slightly different messages will have the same hash.

Uses 
Functions with these properties are used as hash functions for a variety of purposes, not only in cryptography. Practical applications include message integrity checks, digital signatures, authentication, and various information security applications.

A hash function takes a string of any length as input and produces a fixed length string which acts as a kind of "signature" for the data provided. In this way, a person knowing the "hash value" is unable to know the original message, but only the person who knows the original message can prove the "hash value" is created from that message.

A cryptographic hash function should behave as much as possible like a random function while still being deterministic and efficiently computable. A cryptographic hash function is considered "insecure" from a cryptographic point of view, if either of the following is computationally feasible:
Finding a (previously unseen) message that matches a given hash values.
Finding "collisions", in which two different messages have the same hash value.

An attacker who can find any of the above computations can use them to substitute an authorized message with an unauthorized one.

Ideally, it should be impossible to find two different messages whose digests ("hash values") are similar. Also, one would not want an attacker to be able to learn anything useful about a message from its digest ("hash values"). Of course the attacker learns at least one piece of information, the digest itself, by which the attacker can recognise if the same message occurred  again.

In various standards and applications, the two most commonly used hash functions are MD5 and SHA-1.

In 2005, security defects were identified showing that a possible mathematical weakness might exist, like attacks, and recommending a stronger hash function.

In 2007 the National Institute of Standards and Technology announced a contest to design a hash function which will be given the name SHA-3 and be the subject of a FIPS standard.

Different hash algorithms
 MD5: It was designed by Ronald Rivest in 1991 which replaces the earlier version MD4. It is specified as "RFC 1321" in 1992.
 SHA-1: It was developed as part of a project by the U.S. government
 RIPEMD-160: It stands for "RACE Integrity Primitives Evaluation Message Digest". It was developed by Hans Dobbertin, Antoon Bosselaers, and Bart Preneel at the COSIC research group at the Katholieke Universiteit Leuven in Leuven, Belgium, and it was published in 1996.
 Whirlpool
 SHA-2
 SHA-3
 BLAKE2
 BLAKE3

Related pages 
 Avalanche effect

References

Bibliography
 Bruce Schneier. Applied Cryptography. John Wiley & Sons, 1996. .

Other websites
 Hash'em all! – free online text and file hashing with different algorithms
 The Hash function lounge  – a list of hash functions and known attacks
 Helger Lipmaa's links on hash functions 
 Diagrams explaining cryptographic hash functions
 An Illustrated Guide to Cryptographic Hashes by Steve Friedl
 Cryptanalysis of MD5 and SHA: Time for a New Standard by Bruce Schneier
 Hash collision Q&A
 Attacking hash functions by poisoned messages (construction of multiple sensible Postscript messages with the same hash function) 
 What is a hash function?  from RSA Laboratories
 Password Hashing in PHP  by James McGlinn at the PHP Security Consortium
 The code monkey's guide to cryptographic hashes  by Val Henson, "in language that any programmer (and even some managers) can understand."

Cryptography