PuTTY's copy and paste mechanism is by default modelled on the Unix xterm application. The X Window System uses a three-button mouse, and the convention in that system is that the left button selects, the right button extends an existing selection, and the middle button pastes.
Windows often only has two mouse buttons, so when run on Windows, PuTTY is configurable. In PuTTY's default configuration (‘Compromise’), the right button pastes, and the middle button (if you have one) extends a selection.
If you have a three-button mouse and you are already used to the xterm arrangement, you can select it using the ‘Action of mouse buttons’ control.
Alternatively, with the ‘Windows’ option selected, the middle button extends, and the right button brings up a context menu (on which one of the options is ‘Paste’). (This context menu is always available by holding down Ctrl and right-clicking, regardless of the setting of this option.)
(When PuTTY itself is running on Unix, it follows the X Window System convention.)