In simple terms, hash functions transform input data of arbitrary size (e.g. text, binary, etc) to fixedlength output (called hash value, hash code, fingerprint, message digest or simply hash) in a deterministic way. What else?
Types of hash functions:

dedicated functions  specially designed for hashing (e.g. MD4 family including MD4, MD5, SHA1, SHA2, RIPEMD)

based on block cipher  constructed using blockcipher chaining algorithms (e.g. Whirlpool)

based on modular arithmetic  as compression function (e.g. MASH1) but they were proven to be insecure and slow
Security of hash functions:

adhoc basis  very efficient functions where bits of the message are mixed together (bitwise/modular operations, compression, etc) to produce the hash but security is very hard to prove or disprove (a conjecture)

provable secure  based on hard mathematical problems (discrete log, integer factorization, etc) and finding collision is as hard as breaking the underlying problem but they are less efficient to be used in practice
A more academic functional classification:

modification detection codes (MDCs)  unkeyed (only data as input) hash functions used for message integrity

message authentication codes (MACs)  keyed (two inputs: data and a key) hash functions that are used to guarantee both message integrity and message authenticity
Let's have a look at properties/applications of MDCs, leaving MACs for another post.
General properties
Given the above definition, hash functions have at least the following 4 properties:
1. Compression
This is simple to see since any hash functions transform an arbitrary size data (e.g. long text file with thousands of words) into fixed length hash value (e.g. 256 bit) by either mixing the input bits or splitting input into fixedsize blocks.
2. Deterministic
Again, this should be selfevident: given the same input data, the hash function has to return the same hash value.
3. Efficient
Easy to compute the hash value for any given message. It has to be computationally efficient and executed in polynomialtime, please see BigO notation, complexity post for details about algorithms and complexities.
4. Pseudorandomness
Hard to distinguish a hash function from a Random Oracle model.
Security properties
On the other hand, cryptographic hash functions (or oneway encryption) are required to satisfy the following security requirements:
1. Preimage resistance (onewayness)
Definition: given only h(x), it is infeasible to find x. This is the oneway property of the hash functions, it is easy to compute oneway and infeasible to invert.
Application: password storage  any computer system stores the hash of the password instead of plain text password. Let's keep it simple and ignore salting and rainbow table for now.
Intuition: preimage resistance is required, otherwise an adversarial will be able to invert the hashing algorithm and figure out the original passwords.
2. Collision resistance
2.1 Weak collision resistance (second preimage resistance)
Definition: given x and thus h(x), it is infeasible to find another x' where x' != x and h(x') == h(x). On the other hand, due to Pigeonhole principle (or Dirichlet's drawer principle) collisions always exist, the question is how hard is to find them.
Application: file integrity  to check whether the downloaded file has been tampered with or not
Intuition:

preimage resistance is not needed in this case since both file content and hash value are public

second preimage resistance is required to prevent an attacker to modify the file in such a way (to basically find x') that original hash value is preserved
2.2 Strong collision resistance (collision resistance)
Definition: infeasible to find any x1, x2 pair where x1 != x2 and h(x1) == h(x2). See Birthday paradox.
Application: dice roll  players bet (commit) on an output value, one rolls the dice then players reveal their bets.
Intuition:

preimage resistance is required, otherwise another player can invert the hash and find the original bet x

collision resistance is required to prevent the player to reveal another bet x2 that has the same hash value and wins the game
Note: sometimes the terms preimage resistance, second preimage resistance and, collision resistance are confusing and it is important to understand that:

second preimage resistance does not guarantee preimage resistance  you might find another x' that hashes to same hash value (collision) but you can't find the original x

collision resistance implies second preimage resistance  if you can find a pair x1, x2 you can also find x'. Strong resistance implies weak resistance.
3. Nonmalleability
Definition: given only h(x), it is infeasible to find h(y) where and x, y are related in some way (e.g. y=x+1).
Application: auctions  where each participant commits to a value and the highest bidder wins.
Intuition:

preimage resistance and collision resistance are required for the same reasons as above: to prevent finding x and x2

nonmalleability is required to prevent another player to bet more (find and reveal h(y)) even if the original bet x is not known yet
Additional properties
1. Noncorrelation
Inputs bits and output bits should not be correlated in any way.
2. Nearcollision resistance
Infeasible to find any x1, x2 pair such that h(x1) and h(x2) differ in only small number of bits.
3. Partial preimage resistance
It should be difficult to recover any subset (aka substring) of input data.
A bit of history

Message Digest (MD)  developed by Ronald Rivest (except the MD2)

MD2  old, designed for systems with limited memory  BROKEN

MD4  similar to MD2 but adapted for software  BROKEN

MD5  still in use today but it has many weaknesses firstly demonstrated by Hans Dobbertin  BROKEN

MD6  the latest message digest algorithm


Secure Hash Algorithm (SHA)  developed by NIST

SHA 1  deprecated but still widely used  BROKEN

SHA 2  based on the cryptographic concept called "Merkleâ€“Damgard construction"

SHA 3  Keccak hashes family, which are based on the cryptographic concept "sponge construction". NIST held a SHA3 competition and the winner was Keccak


BLAKE  one of the finalists of NIST SHA3 competition, less used in practice

RACE Integrity Primitives Evaluation Message Digest (RIPEMD)  with variants RIPEMD160, RIPEMD256

Tiger  designed to run on 6bit processors and replace MD4, MD5 and SHA1, weaknesses found

Whirlpool  designed by V. Rijmen (coinventor of AES) recommended by New European Schemes for Signatures, Integrity, and Encryption (NESSIE)

SM3  Chinese Standard is 256bit cryptographic hash derived from SHA2

GOST  the Russian national standard