You have to press [I] while holding your mouse on whatever you want to make a key frame on. The reason that it's like this is because almost anything in blender is animatable. even the buttons. This gives for lots of options and creative possibilities, but makes things a little different from other software.
So if you want to add a keyframe for an object, you select the object, and hold your mouse over the 3d view and press [I]. Alternatively you can go to Object > Animation > Insert Keyframe in the 3d viewport.
If you want to add keyframes for an armature, make sure you are in Pose mode of that armature, select the bones you want keyframes for and press [I] while your mouse is hovering the 3D view.
You can also animate settings, right click or press [I] over almost any button in blender and there is an option [+ Insert Keyframe] available so this can be added.
Bonus: This means you can also animate an object by hovering the transform location in the toolbar panel and inserting a keyframe directly from these buttons.