Beta support for the Wii official online multiplayer
This is a feature that has been coming for a long time. More than two years ago, Matthew Parlane and Shawn Hoffman started working on emulating the Wii Wi-Fi networking API in order to run Wii online multiplayer games inside Dolphin. While it was not an easy change in itself, it also required a lot of modifications to core components of the emulator to be implemented properly (for example asynchronous IPC HLE) and a lot of debugging. It is now working well enough that we are releasing it to the public as a beta: don't expect everything to work, but popular games like Mario Kart Wii or Super Smash Bros Brawl can be played online right now.
Alpha support for ARM/Android
About two years ago, Ryan Houdek began the implementation of an ARM port of Dolphin, designed to run on powerful mobile phones and other ARM devices in the future. After a long time spent making Dolphin work well on both ARM and x86, Dolphin can now emulate GameCube and Wii games on recent Android phones. This support is still in early alpha stages: crashes happen, it's slow on Qualcomm hardware because of graphics drivers issues, and it is still missing a ton of features. Nowadays, Dolphin on Android is a two man project: Mathew Maidment is helping Ryan with the UI and making the Android version actually usable.
Global User directory on Windows
This is not exactly a major feature, but it is a big change in how Dolphin works on Windows, and requires user interaction to move from the old configuration system to the new one. Before 4.0, Dolphin configuration was stored next to Dolphin.exe, often causing issues when upgrading to a new version of Dolphin. New versions of Dolphin use a centralized location to store the configuration for all builds, usually My Documents\Dolphin Emulator. The documentation article linked above explains the details of this move, as well as what you need to do to migrate your old configuration to the new system (if you were using Dolphin before).
New AX DSP HLE emulation code
DSP HLE is the main audio emulation technique used in Dolphin. Before 4.0, it was extremely inaccurate and full of bugs, mostly due to how it was implemented. Dolphin 4.0 introduces a full rewrite of the audio emulation used in 99% of games, fixing hundreds of audio related bugs in Dolphin. On the flip side, it is now required to run a game at full speed to get full speed audio out of it, which is a direct consequence of fixing these bugs.