PuTTY has three user-interface actions which can be configured to paste into the terminal (not counting menu items). You can click whichever mouse button (if any) is configured to paste (see section 4.11.1); you can press Shift-Ins; or you can press Ctrl-Shift-V, although that action is not enabled by default.
You can configure which of the available clipboards each of these actions pastes from (including turning the paste action off completely). On platforms with a single system clipboard (such as Windows), the available options are to paste from that clipboard or to paste from PuTTY's internal memory of the last selected text within that window. On X, the standard options are CLIPBOARD or PRIMARY.
(PRIMARY is conceptually similar in that it also refers to the last selected text – just across all applications instead of just this window.)
The two keyboard options each come with a corresponding key to copy to the same clipboard. Whatever you configure Shift-Ins to paste from, Ctrl-Ins will copy to the same location; similarly, Ctrl-Shift-C will copy to whatever Ctrl-Shift-V pastes from.
On X, you can also enter a selection name of your choice. For example, there is a rarely-used standard selection called ‘SECONDARY’, which Emacs (for example) can work with if you hold down the Meta key while dragging to select or clicking to paste; if you configure a PuTTY keyboard action to access this clipboard, then you can interoperate with other applications' use of it. Another thing you could do would be to invent a clipboard name yourself, to create a special clipboard shared only between instances of PuTTY, or between just instances configured in that particular way.