Every time I render an image and try to save it, it just disappears. How do you exactly save a render? I am using a mac.
If you render a single frame by pressing F12, it won't be saved by default. It will occupy one of the render result slots, untill you render a frame again. In the UV/Image Editor, where you can see the rendered result, you can easily save it by going to Image > Save As Image, or by pressing F3:
If by disappears you mean it gets back to 3d view then you should be able to switch the view back to image editor and open last render result.
Otherwise you can also use Output field in your rendering settings to save the render in wanted location. (But as Tardis noted in a comment, you need to use rendering as animation and set frame range to a single frame to output the image to the location.)
As other answers mentioned, Blender doesn't save stills by default, but you can change that if you're alright with writing a tiny bit of python (barely writing any).
If you right-click the "Render" button and select "Edit Source" you can then switch to the text editor, and in the files menu (the button next to "Templates" drop down) you can see properties_render.py file there. When you click it, a cursor will be at the line "defining" the button in the GUI.
At the end of the line, just append
.write_still = True.
If you do that, it will automatically save stills to the output directory you set (the same directory where the rendered animation frames are saved to as well).
Note: If you're on Windows and have Blender installed to the default location (Program Files, usually), you may have to run it as administrator in order to edit the file. Also note, that the safest way to apply the change is to restart Blender after saving the file (Alt + S, or just use the Text dropdown menu and select the Save item).
Congrats! Now your Blender installation will automatically save stills by default to the output directory you set for the project.