NEM has three versions of wallets that can be run locally and independently of an NIS. The NCC is the first wallet and comes equipped with APIs. Open-source Android and iOS mobile apps have been built and are released on the testnet for full testing and will be coming to the App Store and the Google Play store soon. And lastly there is the new lightwallet.
Run with NIS
Install and sync the NIS and then go to http://127.0.0.1:7778/ in your web browser.
Download and Run the Stand-Alone
You can download the lightwallet here: http://bob.nem.ninja/lightwallet-standalone-1.8-cdn-nginx1.8.zip
The password to extract it is: nem
For android: (In Progress...)