Cryptographic hash algorithms
What are hash algorithms ?
A cryptographic hash function is an algorithm (sequence of mathematic and cryptographic operations) that takes an arbitrary amount of data and produces a fixedsize output of enciphered text called a hash. They are usually used to secure an information transfer between two computer system. Hash functions allow mainly to confirm some properties of an IT message, such as its destination, its origin, or just to make the difference between two similar items without revealing its content.
Example of hash function
md5(a) = 0cc175b9c0f1b6a831c399e269772661
md5(b) = 92eb5ffee6ae2fec3ad71c777531578f
You can check that the first requirement is respect, which is determinism, by going yourself on a website which calculate a SHA256 hash. Each SHA256 implementation is identic so you can go on every website or through every programming language you should get the same result than I. Be careful with the interlaced oe which is one unique character. You will observe while the efficiency requirement is respected since the result is instant.
Finally, you will notice that we can easily differenciate the hash of two close words. Of course, checking the security requirement of attack resistance and unicity of the hash is not easy. We need to do that in order to do very hard mathematical operations and even sometimes spend years to make a bit of progress. This is not a task for everyone, and we will have to trust cryptography specialists. Concerning SHA256 the function seams secured until now. But its predecessors where all break by exceptional mathematicians and cryptographs, so we need to keep in mind that this may append to SHA256.
Therefore, many cryptographs often set new alternative functions, modulated to improve some points, which can be a security improvement or a speed improvement for example. The SHA3 function, also called Keccak, and the Whirlpool are example among others of new functions which are set to explore new way of functioning to get better results.
How do they work ?
It is both very simple and very complicated.
Considering the mathematic aspect, everything depends on the hash function which is used: there are dozens of them, some more secured than others. Every existing function is based on different protocols: substitution, permutation, content addition, difference of content… Each function leads in consequence to different results.
To keep it simple, the mathematic aspect of a hash function is not very different from antic cyphertext, or the Enigma code. To understand what the result means or does; we have to know the procedure which was used.
Hash functions have however in common the fact that they are used in a computer science context. Another difference with the previous cryptographic functions is that the objective is not to recover the initial message from the result.
In order to do that, hash functions must follow some rules:

Determinism: the result of the function must be invariable in each circumstance. Whenever the function is used, by whom and independantly of how many times is it used, if the initial message is the same the result must be the same.

Efficiency: The hash function must allow to get an instant result, otherwise it would slow down the whole system and each computer system depending on it.

Attack resistance: two messages which are close must generate two easily separable hash.

Result unicity: It must be impossible to generate two similar results with two different inputs.

MD5
MD5, meaning Message Digest 5, is the director successor of MD4. It was created in 1991 by Ronald Rivest by improving MD4’s architecture in order to counter the potential attacks that could be done onto it. The main problem with this function was the high probability of finding a collision, which was lowered with its successor MD5, but unfortunately did not suppress this risk.
Learn more 
SHA (Secure Hash Algorithm)
SHA1 (Secure Hash Algorithm) is a hash function made by the NSA (National Security Agency) and publicated by the NIST (National Institute of Standards and Technology) as information processing standard. Since 2005, SHA1 has not been considered secure against wellfunded opponents. In 2011, NIST formally deprecated use of SHA1 and disallowed
Learn more 
Blake2
The first Blake version was released around 2008 and was created for a hash function competition. Blake made it to the final round but lost against Keccak witch was chosed to build SHA3. It is a pretty fast function (faster than md5, SHA2, SHA3) on ARM architectures. It is nowadays used by many for exemple Argon2 which won the Password Hashing Competition uses Blake2, RAR file also uses blake2 in some situations.
Learn more 
Whirlpool
Whirlpool was designed by Vincent Rijmen (cocreator of the Advanced Encryption Standard) and Paulo S. L. M. Barreto, who described it in 2000. It has been adopted by the International Organization for Standardization (ISO) and the International Electrotechnical Commission (IEC) as ISO/IEC 101183 international standard. This algorithm is free to use and there are no patent limits.
The purpose of this function is to take for input a message of size less or equal than and to return a digest message of size 512 bits (the « hash ») using a function like the one used in AES (symmetric encryption algorithm).
Learn more