The most widely used proof-of-work schemes are SHA-256, which was introduced by Bitcoin, and scrypt, which is used by currencies such as Litecoin. Some cryptocurrencies, such as Peercoin, use a combined proof-of-work/proof-of-stake scheme (http://en.wikipedia.org/wiki/Cryptocurrency).
SHA-2 is a set of cryptographic hash functions (SHA-224, SHA-256, SHA-384, SHA-512, SHA-512/224, SHA-512/256) designed by the U.S. National Security Agency (NSA) and published in 2001 by the NIST as a U.S. Federal Information Processing Standard (FIPS). SHA stands for Secure Hash Algorithm. SHA-2 includes a significant number of changes from its predecessor, SHA-1. SHA-2 currently consists of a set of six hash functions with digests that are 224, 256, 384 or 512 bits (http://en.wikipedia.org/wiki/SHA-2).
In cryptography, scrypt is a password-based key derivation function created by Colin Percival, originally for the Tarsnap online backup service. The algorithm was specifically designed to make it costly to perform large-scale custom hardware attacks by requiring large amounts of memory. In 2012, the scrypt algorithm was published by IETF as an Internet Draft, intended to become an informational RFC, which has since expired. A simplified version of scrypt is used as a proof-of-work scheme by a number of cryptocurrencies, such as Litecoin and Dogecoin (http://en.wikipedia.org/wiki/Scrypt).
A proof-of-work (POW) system (or protocol, or function) is an economic measure to deter denial of service attacks and other service abuses such as spam on a network by requiring some work from the service requester, usually meaning processing time by a computer. A key feature of these schemes is their asymmetry: the work must be moderately hard (but feasible) on the requester side but easy to check for the service provider. This idea is also known as a CPU cost function, client puzzle, computational puzzle or CPU pricing function. It is distinct from a CAPTCHA, which is intended for a human to solve quickly, rather than a computer (http://en.wikipedia.org/wiki/Proof-of-work).
Proof-of-stake is a method of securing a cryptocurrency network through requesting users to show ownership of a certain amount of currency. It is different from proof-of-work systems that run hashing algorithms to validate electronic transactions. It is most commonly used as a supplement to proof-of-work in Peercoin and a few other electronic currencies (http://en.wikipedia.org/wiki/Proof-of-stake).