This is a free download available for Android 2.1 or higher and available in the Android Marketplace.
Connect to remote servers, exchange files on your Android phone
This app is indispensable for hardcore IT nerds that want to stay on top of managing their computers remotely. ConnectBot is a powerful open-source secure shell (SSH) client that can manage simultaneous SSH sessions, create secure tunnels and copy and paste between other applications.
ConnectBot is available for all versions of Android and is available for free in the Android Marketplace. iOS users, check out iSSH for similar functionality.
Perfect for the occasional mad scientist
The Electronics Toolkit includes a 555 timer and allows users to calculate LED resistor values and more. If this strikes your fancy, the Electrical Toolkit, written by the same author, provides a host of resources and calculators to ensure your electronics project works like a charm.
Both apps are free for iOS and are available in the App Store. Android users can check out ElectroDroid for similar functionality to these two apps.
Amateur Radio Exam Prep
Listen for ET and make new friends around the world
Ever since seeing "Contact" and watching Jodie Foster's character speaking into her radio, I've been a sucker for amateur radio. This is a fantastic app to help you prepare to get your license and start surfing the airwaves. The creator of this app has created apps for each of the licensing levels (Technician, General and Extra) so you can pick whichever is right for you. Morse code is no longer required to become an amateur radio operator, so this is a great time to pick up a new hobby.
Each iOS app is available in the App Store for $4.99. Android users can download Ham Radio Study in the Android Marketplace.