Parity Ethereum is a software stack that lets you run blockchains based on the Ethereum Virtual Machine (EVM) with a number of different consensus engines. That is, it can serve as a client for the public Ethereum network as well as be the platform for you.
Parity is written from the ground-up for correctness-verifiability, modularisation, low-footprint and high-performance. To this end it utilises the Rust language, a hybrid imperative/OO/functional language with an emphasis on efficiency. It is professionally developed by Parity Technologies; we aim to have all important logic 100% unit-tested, all public APIs 100% documented, all code reviewed by multiple peers and follow a pipelined 6-ish-week release cycle similar to the Rust compiler.
For private chain setups, Parity supports a Proof-of-Authority (PoA) consensus engine to be used as a replacement for the Proof-of-Work (PoW) consensus engine. Proof-of-Authority does not depend on nodes solving arbitrarily difficult mathematical problems, but instead uses a hard-configured set of "authorities" - nodes that are explicitly allowed to create new blocks and secure the blockchain. This makes it easier to maintain a private chain and keep the block issuers accountable.
To allow for frictionless development and demos, Parity Ethereum Dev uses a development block making engine instead of PoW. When used, transactions will be instantly "mined" (after being confirmed by Secure Transaction Signer) and all blocks will be accepted. Furthermore, it exposes the standard and Parity specific Ethereum RPC methods.