Screenshots are automatically saved to your Mac desktop as .png files and named "Screenshot date at time .png," where "date" is the current date and "time" is the current time.
If you want to save screenshots to the clipboard instead, press and hold the Control key in addition to the screenshot command keys. So, to save a screenshot of the whole screen to the clipboard, press Control + ⌘ + Shift + 3.
There are, of course, third party utilities that can help you change where screenshots are saved too, but you can also do it using the Terminal. Don't worry, it's very easy. Here's how to change your Mac's screenshot location: