The Base Commerce C SDK is a pre-built library that implements the Base Commerce API (which integrates Credit Card Processing, ACH processing, Secured Vault, and a Push Notifications Platform). The Base Commerce SDKs are available in other languages as well. This allows developers to drop the library into applications and start communicating with the Base Commerce platform.
Without the dot, "cardio" is Rule #1 in Zombieland. With the dot, card.io is PayPal's latest acquisition: the startup formerly known as Lumber Labs, whose technology allows developers to integrate credit card payments into any iOS or Android app with no additional hardware.
PayPal has officially released PayPal X, its next generation platform that includes a variety of APIs and other resources for developers. Primarily aimed at allowing developers to embed payment processing in a wider array of applications and environments, this is a major expansion of PayPal's APIs.