Activation is the process by which a new (or newly restored) iPhone or iPod touch can get by the "Emergency Call Screen" (iPhone) or "Connect to iTunes" screen (not to be confused with Recovery Mode; the activation screen has a battery icon in the top right corner to indicate this) to access the SpringBoard.
The code in charge of this resides in lockdownd, which is always running on iOS and monitors the activation status of the device. Lockdownd patches (which requires a jailbreak whereby a patched kernel can be booted by iBoot without dynamic libraries dynamically patching in RAM) activate your phone and remove the need to activate legitimately through iTunes with an official carrier (this process is also called "hacktivation"), however the iPhone cannot be used to communicate unless an unlock is found for the baseband. Lockdownd patches are only used on the iPhone as the iPod touch has never been denied activation regardless of firmware, country etc.
Activation is handled by https://albert.apple.com/deviceservices/deviceActivation
iTunes generates an Activation Token and sends it to Apple's activation server. Once the Activation Token is validated, the server will generate a WildcardTicket and signs it with Apple's private key. iTunes then calls AMDeviceActivate with the WildcardTicket; The device gets the WildcardTicket and checks if the signature matches. If it does, it get pasts the emergency call screen and allowing the use of the iPhone. All devices actually go through this process. The activation process is outlined in detail in US patent no. 2009/0061934.
Although the iPod touch can be "activated" without an internet connection, some services such as YouTube and Push Notifications will fail to work due to not having a valid authentication token (iPad and iPod touch: Unable to use YouTube or Push notifications Archived 2010-08-11 at the Wayback Machine) so connecting to iTunes will activate the iPod touch fully.
The iPhone needs a cellular data connection for the first time, after the activation in iTunes. You can make calls if an alert says "iPhone is activated". If you don't have a cellular data connection (3G, EDGE, GPRS) you won't be able to make calls and you have only 1 bar of reception. If you only have 1 bar and no carrier at the status bar, it isn't activated correctly.
SAM (Subscriber Artificial Module) can simulate official activation for hacktivated devices.