Carthage is a dependency manager for Swift and iOS that emphasizes simplicity, unlike CocoaPods which creates an entirely new workspace. To install dependencies with Carthage, create a Cartfile specifying the libraries to be installed and run `carthage update --platform iOS`. The frameworks are then manually linked into Xcode by dragging them from the Carthage folder over to Xcode. With Carthage, you need to integrate the frameworks into your project manually, unlike CocoaPods which creates an entirely new workspace. In this tutorial, the author used Carthage to manage dependencies for a Swift and iOS app that displays a recent picture taken on Mars using NASA's API with Alamofire and SwiftyJSON. The author installed the required libraries using Carthage, linked them into Xcode manually, and sent a GET request to the Mars Rover API to receive an image URL which was then loaded in a UIImageView.