Bitcoin Core

Bitcoin Core (formerly Bitcoin-Qt) is the third Bitcoin client, developed by Wladimir van der Laan based on the original reference code by Satoshi Nakamoto.^[1]^[2] It has been bundled with bitcoind since version 0.5. Bitcoin-Qt has been rebranded to Bitcoin Core since version 0.9.0 ^[3].

Bitcoin Core can be used as a desktop client for regular payments or as a server utility for merchants and other payment services.

Source code (and build instructions for supported platforms) can be found on the Bitcoin GitHub page.

Features

Most popular software implementation of a bitcoin full node. Provides trustless validation that all of bitcoin's consensus rules are being followed.
Has an RPC interface allowing developers to interface with Core and access the bitcoin currency trustlessly.
Has a GUI frontend, Bitcoin-Qt, allowing ordinary users to use bitcoin with full validation.
Compatibility with Linux (both GNOME and KDE), Mac OS X and Windows
All functionality of the original wxWidgets client
Asks for confirmation before sending coins
CSV export of transactions
Clearer transaction list with status icons and real-time filtering
Progress bar on initial block download
Languages: Dutch, English, German, Chinese and many more. Translations are being done by volunteers on Transifex.
Sendmany support in UI (send to multiple recipients in one transaction)
Multiple unit support, can show subdivided bitcoins (mBTC, µBTC) for users that like large numbers (only decimal units)
Splash screen that details progress
Debug window
Payment requests (BIP 70)
Coin control