DISCLAIMER This post reflects my personal opinion only; it does not constitute investment advice of any kind.
The Augur project is an Ethereum-based platform for oracles. Oracles are sources of external information for smart contracts.
For example, if you wanted to write a smart contract specifying that account X should transfer some ETH to account Y in case it rains tomorrow, you would need an oracle to provide the weather information for your contract.
Basically, Augur provides smart contracts with economic incentives for people to providing reliable information about the external world.
In addition to that, it is also a prediction market platform because you can also provide information on future events, which effectively amounts to people compting with each other on who best correctly predicts future information. This enables betting services on anything you can imagine.
All of that in a decentralized way, with no need to trust any third party and outside government intervention.
REP stands for reputation. It's the token used by who wants to create events to be betted on or by whoever wants to help predict the odds of event outcomes.
Gnosis: Another Ethereum-based prediction market platform.
Stox: Prediction market platform, run on top of Ethereum using Bancor contracts.
Hivemind: Oracle protocol. Formerly known as Truthcoin.
Bodhi-network: Prediction markets running on top of QTUM.