By default, copy and paste works similarly to the X Window System. You use the left mouse button to select text in the PuTTY window. The act of selection automatically copies the text to the clipboard: there is no need to press Ctrl-Ins or Ctrl-C or anything else. In fact, pressing Ctrl-C will send a Ctrl-C character to the other end of your connection (just like it does the rest of the time), which may have unpleasant effects. The only thing you need to do, to copy text to the clipboard, is to select it.
This default dates from the days when PuTTY was new, and its main target audience was ‘Unix expatriates’ – people who mostly used Unix desktop systems and were used to the X copy/paste mechanism, having to use Windows instead. The aim was ‘closest thing you can get to an xterm on Windows’.
If you prefer a more ‘Windows native’ interface for copy and paste, via keystrokes or menu actions, with nothing happening automatically on select, then you can reconfigure the behaviour flexibly in PuTTY's configuration. See section 4.11 for details.
To paste the clipboard contents into a PuTTY window, by default you click the right mouse button. If you have a three-button mouse and are used to X applications, you can configure pasting to be done by the middle button instead, but this is not the default because most Windows users don't have a middle button at all.
You can also paste by pressing Shift-Ins.